JEE MAIN · MATHEMATICS · DEFINITE INTEGRATION
JEE Main Definite Integration Questions & Solutions
216 solved questions on Definite Integration, ranging from easy to JEE-Advanced-flavour hard. Click any to see the full solution.
216 solved questions on Definite Integration, ranging from easy to JEE-Advanced-flavour hard. Click any to see the full solution.
Let $f$ be $a$ differentiable function defined on $\left[ {0,{\pi \over 2}} \right]$ such that $f(x) 0$ and $$f(x) + \int_0^x {f(t)\sqrt {1 - {{({{\log }_e}f(t))}^2}} dt =…
View solution →Let $f(x) = \int\limits_0^x {{e^t}f(t)dt + {e^x}}$ be a differentiable function for all x$\in$R. Then f(x) equals :
View solution →The value of $$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\sum\limits_{j = 1}^n {{{(2j - 1) + 8n} \over {(2j - 1) + 4n}}} $$ is equal to :
View solution →If $$\int_\limits{\frac{\pi}{6}}^{\frac{\pi}{3}} \sqrt{1-\sin 2 x} d x=\alpha+\beta \sqrt{2}+\gamma \sqrt{3}$$, where $\alpha, \beta$ and $\gamma$ are rational numbers, then $3…
View solution →$$\lim _\limits{x \rightarrow \frac{\pi}{2}}\left(\frac{\int_{x^3}^{(\pi / 2)^3}\left(\sin \left(2 t^{1 / 3}\right)+\cos \left(t^{1 / 3}\right)\right) d…
View solution →$$\int\limits_{{{3\sqrt 2 } \over 4}}^{{{3\sqrt 3 } \over 4}} {{{48} \over {\sqrt {9 - 4{x^2}} }}dx} $$ is equal to :
View solution →If $$f(t)=\int_\limits0^\pi \frac{2 x \mathrm{~d} x}{1-\cos ^2 \mathrm{t} \sin ^2 x}, 0
View solution →Let $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x)=a e^{2 x}+b e^x+c x$. If $f(0)=-1, f^{\prime}\left(\log _e 2\right)=21$ and $\int_0^{\log _e 4}(f(x)-c x) d…
View solution →Let $f, g:(0, \infty) \rightarrow \mathbb{R}$ be two functions defined by $f(x)=\int\limits_{-x}^x\left(|t|-t^2\right) e^{-t^2} d t$ and $g(x)=\int\limits_0^{x^2} t^{1 / 2} e^{-t}…
View solution →The integral $\int\limits_{0}^{\frac{\pi}{2}} \frac{1}{3+2 \sin x+\cos x} \mathrm{~d} x$ is equal to :
View solution →Let $$\lim _\limits{n \rightarrow \infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2 n}{\left(n^2+1\right) \sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8 n}{\left(n^2+4\right)…
View solution →Let [t] denote the greatest integer less than or equal to t. Then the value of $\int\limits_1^2 {\left| {2x - \left[ {3x} \right]} \right|dx}$ is ______.
View solution →The value of the integral $\int\limits_{1/2}^2 {{{{{\tan }^{ - 1}}x} \over x}dx}$ is equal to :
View solution →If [x] is the greatest integer $\le$ x, then $${\pi ^2}\int\limits_0^2 {\left( {\sin {{\pi x} \over 2}} \right)(x - [x]} {)^{[x]}}dx$$ is equal to :
View solution →$$\mathop {\lim }\limits_{x \to 1} \left( {{{\int\limits_0^{{{\left( {x - 1} \right)}^2}} {t\cos \left( {{t^2}} \right)dt} } \over {\left( {x - 1} \right)\sin \left( {x - 1}…
View solution →Let $f:[0,\infty ) \to [0,\infty )$ be defined as $f(x) = \int_0^x {[y]dy}$where [x] is the greatest integer less than or equal to x. Which of the following is true?
View solution →Let f : (a, b) $\to$ R be twice differentiable function such that $f(x) = \int_a^x {g(t)dt}$ for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a,…
View solution →The value of the integral $\int_1^2 {\left( {{{{t^4} + 1} \over {{t^6} + 1}}} \right)dt}$ is
View solution →If $$\int_0^\pi {({{\sin }^3}x){e^{ - {{\sin }^2}x}}dx = \alpha - {\beta \over e}\int_0^1 {\sqrt t {e^t}dt} } $$, then $\alpha$ + $\beta$ is equal to ____________.
View solution →Let f(x) be a differentiable function defined on [0, 2] such that f'(x) = f'(2 $-$ x) for all x$\in$ (0, 2), f(0) = 1 and f(2) = e2. Then the value of $\int\limits_0^2 {f(x)} dx$…
View solution →Let $$f(x)=\frac{x}{\left(1+x^{n}\right)^{\frac{1}{n}}}, x \in \mathbb{R}-\{-1\}, n \in \mathbb{N}, n 2$$. If $f^{n}(x)=\left(f \circ f \circ f \ldots .\right.$. upto $n$ times)…
View solution →The integral $${{24} \over \pi }\int_0^{\sqrt 2 } {{{(2 - {x^2})dx} \over {(2 + {x^2})\sqrt {4 + {x^4}} }}} $$ is equal to ____________.
View solution →$$\int\limits_{0}^{2}\left(\left|2 x^{2}-3 x\right|+\left[x-\frac{1}{2}\right]\right) \mathrm{d} x$$, where [t] is the greatest integer function, is equal to :
View solution →$$\mathop {\lim }\limits_{x \to 0} {{\int_0^x {t\sin \left( {10t} \right)dt} } \over x}$$ is equal to
View solution →If [ . ] represents the greatest integer function, then the value of $$\left| {\int\limits_0^{\sqrt {{\pi \over 2}} } {\left[ {[{x^2}] - \cos x} \right]dx} } \right|$$ is…
View solution →If the value of the integral $$\int\limits_0^5 {{{x + [x]} \over {{e^{x - [x]}}}}dx = \alpha {e^{ - 1}} + \beta } $$, where $\alpha$, $\beta$ $\in$ R, 5$\alpha$ + 6$\beta$ = 0,…
View solution →Let $f(x) = \left| {x - 2} \right|$ and g(x) = f(f(x)), $x \in \left[ {0,4} \right]$. Then $\int\limits_0^3 {\left( {g(x) - f(x)} \right)} dx$ is equal to:
View solution →If $\int\limits_0^{\frac{\pi}{3}} \cos ^4 x \mathrm{~d} x=\mathrm{a} \pi+\mathrm{b} \sqrt{3}$, where $\mathrm{a}$ and $\mathrm{b}$ are rational numbers, then $9 \mathrm{a}+8…
View solution →The value of $$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\sum\limits_{r = 0}^{2n - 1} {{{{n^2}} \over {{n^2} + 4{r^2}}}} $$ is :
View solution →Let P(x) = x2 + bx + c be a quadratic polynomial with real coefficients such that $\int_0^1 {P(x)dx}$ = 1 and P(x) leaves remainder 5 when it is divided by (x $-$ 2). Then the…
View solution →Let $$f_{n}=\int_\limits{0}^{\frac{\pi}{2}}\left(\sum_\limits{k=1}^{n} \sin ^{k-1} x\right)\left(\sum_\limits{k=1}^{n}(2 k-1) \sin ^{k-1} x\right) \cos x d x, n \in \mathbb{N}$$.…
View solution →If [t] denotes the greatest integer $\le \mathrm{t}$, then the value of ${{3(e - 1)} \over e}\int\limits_1^2 {{x^2}{e^{[x] + [{x^3}]}}dx}$ is :
View solution →If the real part of the complex number ${(1 - \cos \theta + 2i\sin \theta )^{ - 1}}$ is ${1 \over 5}$ for $\theta \in (0,\pi )$, then the value of the integral $\int_0^\theta…
View solution →The value of $${{{e^{ - {\pi \over 4}}} + \int\limits_0^{{\pi \over 4}} {{e^{ - x}}{{\tan }^{50}}xdx} } \over {\int\limits_0^{{\pi \over 4}} {{e^{ - x}}({{\tan }^{49}}x + {{\tan…
View solution →Let $\alpha \in (0,1)$ and $\beta = {\log _e}(1 - \alpha )$. Let $${P_n}(x) = x + {{{x^2}} \over 2} + {{{x^3}} \over 3}\, + \,...\, + \,{{{x^n}} \over n},x \in (0,1)$$. Then the…
View solution →The value of $$\int\limits_{{{ - \pi } \over 2}}^{{\pi \over 2}} {{1 \over {1 + {e^{\sin x}}}}dx} $$ is:
View solution →Let $a$ and $b$ be real constants such that the function $f$ defined by $$f(x)=\left\{\begin{array}{ll}x^2+3 x+a & , x \leq 1 \\ b x+2 & , x1\end{array}\right.$$ be differentiable…
View solution →Let ${J_{n,m}} = \int\limits_0^{{1 \over 2}} {{{{x^n}} \over {{x^m} - 1}}dx}$, $\forall$ n > m and n, m $\in$ N. Consider a matrix $A = {[{a_{ij}}]_{3 \times 3}}$ where $${a_{ij}}…
View solution →The value of the integral $\int_\limits{-1}^2 \log _e\left(x+\sqrt{x^2+1}\right) d x$ is
View solution →If the integral $$525 \int_\limits0^{\frac{\pi}{2}} \sin 2 x \cos ^{\frac{11}{2}} x\left(1+\operatorname{Cos}^{\frac{5}{2}} x\right)^{\frac{1}{2}} d x$$ is equal to $(n…
View solution →The integral $16\int\limits_1^2 {{{dx} \over {{x^3}{{\left( {{x^2} + 2} \right)}^2}}}}$ is equal to
View solution →$$\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over {1 + n}} + {1 \over {2 + n}} + {1 \over {3 + n}}\, + \,...\, + \,{1 \over {2n}}} \right]$$ is equal to
View solution →The integral $\int\limits_0^1 {{1 \over {{7^{\left[ {{1 \over x}} \right]}}}}dx}$, where [ . ] denotes the greatest integer function, is equal to
View solution →The integral $$\int_\limits{1 / 4}^{3 / 4} \cos \left(2 \cot ^{-1} \sqrt{\frac{1-x}{1+x}}\right) d x$$ is equal to
View solution →If $$a = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {{{2n} \over {{n^2} + {k^2}}}} $$ and $f(x) = \sqrt {{{1 - \cos x} \over {1 + \cos x}}}$, $x \in (0,1)$, then…
View solution →If $$\mathrm{n}(2 \mathrm{n}+1) \int_{0}^{1}\left(1-x^{\mathrm{n}}\right)^{2 \mathrm{n}} \mathrm{d} x=1177 \int_{0}^{1}\left(1-x^{\mathrm{n}}\right)^{2 \mathrm{n}+1} \mathrm{~d}…
View solution →Let $$f(x)=\int_\limits0^x g(t) \log _{\mathrm{e}}\left(\frac{1-\mathrm{t}}{1+\mathrm{t}}\right) \mathrm{dt}$$, where $g$ is a continuous odd function. If $$\int_{-\pi / 2}^{\pi /…
View solution →$$\lim _\limits{n \rightarrow \infty}\left\{\left(2^{\frac{1}{2}}-2^{\frac{1}{3}}\right)\left(2^{\frac{1}{2}}-2^{\frac{1}{5}}\right) \ldots . .\left(2^{\frac{1}{2}}-2^{\frac{1}{2…
View solution →The value of the definite integral$$\int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{dx} \over {(1 + {e^{x\cos x}})({{\sin }^4}x + {{\cos }^4}x)}}} $$ is equal to :
View solution →If $$\int\limits_0^2 {\left( {\sqrt {2x} - \sqrt {2x - {x^2}} } \right)dx = \int\limits_0^1 {\left( {1 - \sqrt {1 - {y^2}} - {{{y^2}} \over 2}} \right)dy + \int\limits_1^2 {\left(…
View solution →If $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function satisfying $$\int_\limits{0}^{\frac{\pi}{2}} f(\sin 2 x) \sin x d x+\alpha \int_\limits{0}^{\frac{\pi}{4}}…
View solution →Let the domain of the function$$f(x) = {\log _4}\left( {{{\log }_5}\left( {{{\log }_3}(18x - {x^2} - 77)} \right)} \right)$$ be (a, b). Then the value of the integral…
View solution →If ${I_{m,n}} = \int\limits_0^1 {{x^{m - 1}}{{(1 - x)}^{n - 1}}dx}$, for m, $n \ge 1$, and $$\int\limits_0^1 {{{{x^{m - 1}} + {x^{n - 1}}} \over {{{(1 + x)}^{m + 1}}}}} dx =…
View solution →The value of $\alpha$ for which $4\alpha \int\limits_{ - 1}^2 {{e^{ - \alpha \left| x \right|}}dx} = 5$, is:
View solution →$$\mathop {\lim }\limits_{n \to \infty } {1 \over {{2^n}}}\left( {{1 \over {\sqrt {1 - {1 \over {{2^n}}}} }} + {1 \over {\sqrt {1 - {2 \over {{2^n}}}} }} + {1 \over {\sqrt {1 - {3…
View solution →The value of the integral $\int\limits_0^1 {{{\sqrt x dx} \over {(1 + x)(1 + 3x)(3 + x)}}}$ is :
View solution →If the value of the integral $$\int_\limits{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^2 \cos x}{1+\pi^x}+\frac{1+\sin ^2 x}{1+e^{\sin x^{2123}}}\right) d…
View solution →Let $f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathbf{R}$ be a differentiable function such that $f(0)=\frac{1}{2}$. If the $$\lim _\limits{x \rightarrow 0}…
View solution →If $$\int_0^{\frac{\pi}{4}} \frac{\sin ^2 x}{1+\sin x \cos x} \mathrm{~d} x=\frac{1}{\mathrm{a}} \log _{\mathrm{e}}\left(\frac{\mathrm{a}}{3}\right)+\frac{\pi}{\mathrm{b}…
View solution →If $$f(x) = \left\{ {\matrix{ {\int\limits_0^x {\left( {5 + \left| {1 - t} \right|} \right)dt,} } & {x > 2} \cr {5x + 1,} & {x \le 2} \cr } } \right.$$, then
View solution →If $$\int\limits_{ - a}^a {\left( {\left| x \right| + \left| {x - 2} \right|} \right)} dx = 22$$, (a > 2) and [x] denotes the greatest integer $\le$ x, then$\int\limits_{ - a}^a…
View solution →$$\int_0^5 {\cos \left( {\pi \left( {x - \left[ {{x \over 2}} \right]} \right)} \right)dx} $$, where [t] denotes greatest integer less than or equal to t, is equal to:
View solution →$$\mathop {\lim }\limits_{n \to \infty } \left( {{{{n^2}} \over {({n^2} + 1)(n + 1)}} + {{{n^2}} \over {({n^2} + 4)(n + 2)}} + {{{n^2}} \over {({n^2} + 9)(n + 3)}} + \,\,....\,\,…
View solution →The value of $$\lim _\limits{n \rightarrow \infty} \sum_\limits{k=1}^n \frac{n^3}{\left(n^2+k^2\right)\left(n^2+3 k^2\right)}$$ is :
View solution →If I1 = $\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{100}}} dx$ and I2 = $\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{101}}} dx$ such that I2 = $\alpha$I1 then…
View solution →Let a function $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as : $$f(x)= \begin{cases}\int\limits_{0}^{x}(5-|t-3|) d t, & x4 \\ x^{2}+b x & , x \leq 4\end{cases}$$ where…
View solution →If ${b_n} = \int_0^{{\pi \over 2}} {{{{{\cos }^2}nx} \over {\sin x}}dx,\,n \in N}$, then
View solution →If m and n respectively are the number of local maximum and local minimum points of the function $f(x) = \int\limits_0^{{x^2}} {{{{t^2} - 5t + 4} \over {2 + {e^t}}}dt}$, then the…
View solution →The value of $$\int\limits_0^\pi {{{{e^{\cos x}}\sin x} \over {(1 + {{\cos }^2}x)({e^{\cos x}} + {e^{ - \cos x}})}}dx} $$ is equal to:
View solution →The value of the integral $$\int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{x + {\pi \over 4}} \over {2 - \cos 2x}}dx} $$ is :
View solution →The value of b 3 for which $$12\int\limits_3^b {{1 \over {({x^2} - 1)({x^2} - 4)}}dx = {{\log }_e}\left( {{{49} \over {40}}} \right)} $$, is equal to ___________.
View solution →Let $\alpha0$. If $\int\limits_0^\alpha \frac{x}{\sqrt{x+\alpha}-\sqrt{x}} \mathrm{~d} x=\frac{16+20 \sqrt{2}}{15}$, then $\alpha$ is equal to :
View solution →The value of $\sum\limits_{n = 1}^{100} {\int\limits_{n - 1}^n {{e^{x - [x]}}dx} }$, where [ x ] is the greatest integer $\le$ x, is :
View solution →Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of the integral $\int_{-3}^{101}\left([\sin (\pi x)]+e^{[\cos (2 \pi x)]}\right) d x$ is equal to
View solution →The value of $$\int\limits_{ - {\pi \over 2}}^{{\pi \over 2}} {\left( {{{1 + {{\sin }^2}x} \over {1 + {\pi ^{\sin x}}}}} \right)} \,dx$$ is
View solution →If the normal to the curve y(x) = $\int\limits_0^x {(2{t^2} - 15t + 10)dt}$ at a point (a, b) is parallel to the line x + 3y = $-$5, a > 1, then the value of | a + 6b | is equal…
View solution →Let [t] denote the greatest integer $\le$ t. Then the value of $8.\int\limits_{ - {1 \over 2}}^1 {([2x] + |x|)dx}$ is ___________.
View solution →The value of $\int_\limits{-\pi}^\pi \frac{2 y(1+\sin y)}{1+\cos ^2 y} d y$ is :
View solution →The value of $\int\limits_{ - 1}^1 {{x^2}{e^{[{x^3}]}}} dx$, where [ t ] denotes the greatest integer $\le$ t, is :
View solution →Let $I=\int_{\pi / 4}^{\pi / 3}\left(\frac{8 \sin x-\sin 2 x}{x}\right) d x$. Then
View solution →Let ${I_n} = \int_1^e {{x^{19}}{{(\log |x|)}^n}} dx$, where n$\in$N. If (20)I10 = $\alpha$I9 + $\beta$I8, for natural numbers $\alpha$ and $\beta$, then $\alpha$ $-$ $\beta$…
View solution →Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined as $f(x)=a \sin \left(\frac{\pi[x]}{2}\right)+[2-x], a \in \mathbb{R}$ where $[t]$ is the greatest integer less…
View solution →Let a be a positive real number such that $\int_0^a {{e^{x - [x]}}} dx = 10e - 9$ where [ x ] is the greatest integer less than or equal to x. Then a is equal to:
View solution →Let f : R $\to$ R be defined as f(x) = e$-$xsinx. If F : [0, 1] $\to$ R is a differentiable function with that F(x) = $\int_0^x {f(t)dt}$, then the value of $\int_0^1 {(F'(x) +…
View solution →The value of the integral $\int\limits_{ - 1}^1 {\log \left( {x + \sqrt {{x^2} + 1} } \right)dx}$ is :
View solution →Let P(x) be a real polynomial of degree 3 which vanishes at x = $-$3. Let P(x) have local minima at x = 1, local maxima at x = $-$1 and $\int\limits_{ - 1}^1 {P(x)dx}$ = 18, then…
View solution →Let f : R $\to$ R be a differentiable function such that $$f\left( {{\pi \over 4}} \right) = \sqrt 2 ,\,f\left( {{\pi \over 2}} \right) = 0$$ and $f'\left( {{\pi \over 2}} \right)…
View solution →If $${U_n} = \left( {1 + {1 \over {{n^2}}}} \right)\left( {1 + {{{2^2}} \over {{n^2}}}} \right)^2.....\left( {1 + {{{n^2}} \over {{n^2}}}} \right)^n$$, then $\mathop {\lim…
View solution →The value of $\int\limits_{ - \pi /2}^{\pi /2} {{{{{\cos }^2}x} \over {1 + {3^x}}}} dx$ is :
View solution →Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f'(x) $\ne$ 0 for all x $\in$ R. If $$\left| {\matrix{ {f(x)} & {f'(x)} \cr {f'(x)} &…
View solution →If $\int\limits_{0}^{1} \frac{1}{\left(5+2 x-2 x^{2}\right)\left(1+e^{(2-4 x)}\right)} d x=\frac{1}{\alpha} \log _{e}\left(\frac{\alpha+1}{\beta}\right), \alpha, \beta0$, then…
View solution →Let $$g(t) = \int_{ - \pi /2}^{\pi /2} {\cos \left( {{\pi \over 4}t + f(x)} \right)} dx$$, where $f(x) = {\log _e}\left( {x + \sqrt {{x^2} + 1} } \right),x \in R$. Then which one…
View solution →Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{x}{\left(1+x^4\right)^{1 / 4}}$, and $g(x)=f(f(f(f(x))))$. Then, $18 \int_0^{\sqrt{2 \sqrt{5}}} x^2…
View solution →Let $$\beta(\mathrm{m}, \mathrm{n})=\int_\limits0^1 x^{\mathrm{m}-1}(1-x)^{\mathrm{n}-1} \mathrm{~d} x, \mathrm{~m}, \mathrm{n}0$$. If $$\int_\limits0^1\left(1-x^{10}\right)^{20}…
View solution →Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{4^x}{4^x+2}$ and $$M=\int_\limits{f(a)}^{f(1-a)} x \sin ^4(x(1-x)) d x,…
View solution →Let $[t]$ denote the largest integer less than or equal to $t$. If $$\int_\limits0^3\left(\left[x^2\right]+\left[\frac{x^2}{2}\right]\right) \mathrm{d} x=\mathrm{a}+\mathrm{b}…
View solution →If the value of the integral $\int\limits_{-1}^1 \frac{\cos \alpha x}{1+3^x} d x$ is $\frac{2}{\pi}$.Then, a value of $\alpha$ is
View solution →For $m, n 0$, let $\alpha(m, n)=\int_\limits{0}^{2} t^{m}(1+3 t)^{n} d t$. If $11 \alpha(10,6)+18 \alpha(11,5)=p(14)^{6}$, then $p$ is equal to ___________.
View solution →$\lim\limits_{n \rightarrow \infty} \frac{3}{n}\left\{4+\left(2+\frac{1}{n}\right)^2+\left(2+\frac{2}{n}\right)^2+\ldots+\left(3-\frac{1}{n}\right)^2\right\}$ is equal to :
View solution →Let the slope of the line $45 x+5 y+3=0$ be $27 r_1+\frac{9 r_2}{2}$ for some $r_1, r_2 \in \mathbb{R}$. Then $$\lim _\limits{x \rightarrow 3}\left(\int_3^x \frac{8 t^2}{\frac{3…
View solution →If ƒ(a + b + 1 - x) = ƒ(x), for all x, where a and b are fixed positive real numbers, then ${1 \over {a + b}}\int_a^b {x\left( {f(x) + f(x + 1)} \right)} dx$ is equal to:
View solution →Let for $x \in \mathbb{R}, S_{0}(x)=x, S_{k}(x)=C_{k} x+k \int_{0}^{x} S_{k-1}(t) d t$, where $C_{0}=1, C_{k}=1-\int_{0}^{1} S_{k-1}(x) d x, k=1,2,3, \ldots$ Then $S_{2}(3)+6…
View solution →$$\left|\frac{120}{\pi^3} \int_\limits0^\pi \frac{x^2 \sin x \cos x}{\sin ^4 x+\cos ^4 x} d x\right| \text { is equal to }$$ ________.
View solution →If $$\int_\limits{0}^{1}\left(x^{21}+x^{14}+x^{7}\right)\left(2 x^{14}+3 x^{7}+6\right)^{1 / 7} d x=\frac{1}{l}(11)^{m / n}$$ where $l, m, n \in \mathbb{N}, m$ and $n$ are coprime…
View solution →Let f(x) and g(x) be two functions satisfying f(x2) + g(4 $-$ x) = 4x3 and g(4 $-$ x) + g(x) = 0, then the value of $\int\limits_{ - 4}^4 {f{{(x)}^2}dx}$ is
View solution →Let f : R $\to$ R be a continuous function such that f(x) + f(x + 1) = 2, for all x$\in$R. If ${I_1} = \int\limits_0^8 {f(x)dx}$ and ${I_2} = \int\limits_{ - 1}^3 {f(x)dx}$, then…
View solution →Let f : (0, 2) $\to$ R be defined as f(x) = log2$\left( {1 + \tan \left( {{{\pi x} \over 4}} \right)} \right)$. Then, $$\mathop {\lim }\limits_{n \to \infty } {2 \over n}\left(…
View solution →The integral $\int_0^\pi \frac{8 x d x}{4 \cos ^2 x+\sin ^2 x}$ is equal to
View solution →The integral $\int\limits_0^2 {\left| {\left| {x - 1} \right| - x} \right|dx}$ is equal to______.
View solution →For x > 0, if $f(x) = \int\limits_1^x {{{{{\log }_e}t} \over {(1 + t)}}dt}$, then $f(e) + f\left( {{1 \over e}} \right)$ is equal to :
View solution →If $\lim\limits _{t \rightarrow 0}\left(\int\limits_0^1(3 x+5)^t d x\right)^{\frac{1}{t}}=\frac{\alpha}{5 e}\left(\frac{8}{5}\right)^{\frac{2}{3}}$, then $\alpha$ is equal to…
View solution →$\int_\limits{0}^{\infty} \frac{6}{e^{3 x}+6 e^{2 x}+11 e^{x}+6} d x=$
View solution →Consider the integral $I = \int_0^{10} {{{[x]{e^{[x]}}} \over {{e^{x - 1}}}}dx}$, where [x] denotes the greatest integer less than or equal to x. Then the value of I is equal to :
View solution →$$\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over n} + {n \over {{{(n + 1)}^2}}} + {n \over {{{(n + 2)}^2}}} + ........ + {n \over {{{(2n + 1)}^2}}}} \right]$$ is equal to…
View solution →The value of the integral $$\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {(1 + {e^x})({{\sin }^6}x + {{\cos }^6}x)}}} $$ is equal to
View solution →The value of the integral $$\int_\limits{-\log _{e} 2}^{\log _{e} 2} e^{x}\left(\log _{e}\left(e^{x}+\sqrt{1+e^{2 x}}\right)\right) d x$$ is equal to :
View solution →Let a function ƒ : [0, 5] $\to$ R be continuous, ƒ(1) = 3 and F be defined as : $F(x) = \int\limits_1^x {{t^2}g(t)dt}$ , where $g(t) = \int\limits_1^t {f(u)du}$ Then for the…
View solution →Among (S1): $\lim_\limits{n \rightarrow \infty} \frac{1}{n^{2}}(2+4+6+\ldots \ldots+2 n)=1$ (S2) : $$\lim_\limits{n \rightarrow \infty}…
View solution →Which of the following statements is correct for the function g($\alpha$) for $\alpha$ $\in$ R such that $$g(\alpha ) = \int\limits_{{\pi \over 6}}^{{\pi \over 3}} {{{{{\sin…
View solution →If the value of the integral $$\int\limits_0^{{1 \over 2}} {{{{x^2}} \over {{{\left( {1 - {x^2}} \right)}^{{3 \over 2}}}}}} dx$$ is ${k \over 6}$, then k is equal to :
View solution →The value of the integral, $\int\limits_1^3 {[{x^2} - 2x - 2]dx}$, where [x] denotes the greatest integer less than or equal to x, is :
View solution →The minimum value of the twice differentiable function $$f(x)=\int\limits_{0}^{x} \mathrm{e}^{x-\mathrm{t}} f^{\prime}(\mathrm{t}) \mathrm{dt}-\left(x^{2}-x+1\right)…
View solution →$$ \begin{aligned} &\text { If } \lim _{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(n k+1)+(n k+2)+\ldots+(n k+n)] \\ &=33 \cdot \lim _{n \rightarrow \infty}…
View solution →If the integral $$\int_0^{10} {{{[\sin 2\pi x]} \over {{e^{x - [x]}}}}} dx = \alpha {e^{ - 1}} + \beta {e^{ - {1 \over 2}}} + \gamma $$, where $\alpha$, $\beta$, $\gamma$ are…
View solution →Let {x} and [x] denote the fractional part of x and the greatest integer $\le$ x respectively of a real number x. If $\int_0^n {\left\{ x \right\}dx} ,\int_0^n {\left[ x…
View solution →$$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{r \over {2{r^2} - 7rn + 6{n^2}}}} $$ is equal to :
View solution →If $$\int\limits_0^{100\pi } {{{{{\sin }^2}x} \over {{e^{\left( {{x \over \pi } - \left[ {{x \over \pi }} \right]} \right)}}}}dx = {{\alpha {\pi ^3}} \over {1 + 4{\pi ^2}}},\alpha…
View solution →Let $f(x)=\int_0^x\left(t+\sin \left(1-e^t\right)\right) d t, x \in \mathbb{R}$. Then, $\lim _\limits{x \rightarrow 0} \frac{f(x)}{x^3}$ is equal to
View solution →Let $F:[3,5] \to R$ be a twice differentiable function on (3, 5) such that $F(x) = {e^{ - x}}\int\limits_3^x {(3{t^2} + 2t + 4F'(t))dt}$. If $F'(4) = {{\alpha {e^\beta } - 224}…
View solution →The integral $\int\limits_1^2 {{e^x}.{x^x}\left( {2 + {{\log }_e}x} \right)} dx$ equals :
View solution →$\int\limits_{ - \pi }^\pi {\left| {\pi - \left| x \right|} \right|dx}$ is equal to :
View solution →Let $5 f(x)+4 f\left(\frac{1}{x}\right)=\frac{1}{x}+3, x 0$. Then $18 \int_\limits{1}^{2} f(x) d x$ is equal to :
View solution →The value of the definite integral $\int\limits_{\pi /24}^{5\pi /24} {{{dx} \over {1 + \root 3 \of {\tan 2x} }}}$ is :
View solution →If $\theta$1 and $\theta$2 be respectively the smallest and the largest values of $\theta$ in (0, 2$\pi$) - {$\pi$} which satisfy the equation, 2cot2$\theta$ - ${5 \over {\sin…
View solution →The value of the integral $\int\limits_{ - 2}^2 {{{|{x^3} + x|} \over {({e^{x|x|}} + 1)}}dx}$ is equal to :
View solution →Let $$f(\theta ) = \sin \theta + \int\limits_{ - \pi /2}^{\pi /2} {(\sin \theta + t\cos \theta )f(t)dt} $$. Then the value of $\left| {\int_0^{\pi /2} {f(\theta )d\theta } }…
View solution →The minimum value of the function $f(x) = \int\limits_0^2 {{e^{|x - t|}}dt}$ is :
View solution →Let $${a_n} = \int\limits_{ - 1}^n {\left( {1 + {x \over 2} + {{{x^2}} \over 3} + \,\,.....\,\, + \,\,{{{x^{n - 1}}} \over n}} \right)dx} $$ for every n $\in$ N. Then the sum of…
View solution →If [x] denotes the greatest integer less than or equal to x, then the value of the integral $\int_{ - \pi /2}^{\pi /2} {[[x] - \sin x]dx}$ is equal to :
View solution →Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If $\int_0^x {\sqrt {1 - {{(f'(t))}^2}} dt = \int_0^x {f(t)dt} }$, $0 \le x \le 1$ and f(0) = 0,…
View solution →$\int\limits_{0}^{20 \pi}(|\sin x|+|\cos x|)^{2} d x \text { is equal to }$
View solution →The integral $\int_0^\pi \frac{(x+3) \sin x}{1+3 \cos ^2 x} d x$ is equal to
View solution →Let $f$ be a twice differentiable function on $\mathbb{R}$. If $f^{\prime}(0)=4$ and $$f(x) + \int\limits_0^x {(x - t)f'(t)dt = \left( {{e^{2x}} + {e^{ - 2x}}} \right)\cos 2x + {2…
View solution →Let $f(x)=2+|x|-|x-1|+|x+1|, x \in \mathbf{R}$. Consider $$(\mathrm{S} 1): f^{\prime}\left(-\frac{3}{2}\right)+f^{\prime}\left(-\frac{1}{2}\right)+f^{\prime}\left(\frac{1}{2}\right…
View solution →The value of $\int\limits_0^{2\pi } {{{x{{\sin }^8}x} \over {{{\sin }^8}x + {{\cos }^8}x}}} dx$ is equal to :
View solution →Suppose f(x) is a polynomial of degree four, having critical points at –1, 0, 1. If T = {x $\in$ R | f(x) = f(0)}, then the sum of squares of all the elements of T is :
View solution →Let $f(x)=\min \{[x-1],[x-2], \ldots,[x-10]\}$ where [t] denotes the greatest integer $\leq \mathrm{t}$. Then $$\int\limits_{0}^{10} f(x) \mathrm{d}…
View solution →Let $$\mathop {Max}\limits_{0\, \le x\, \le 2} \left\{ {{{9 - {x^2}} \over {5 - x}}} \right\} = \alpha $$ and $$\mathop {Min}\limits_{0\, \le x\, \le 2} \left\{ {{{9 - {x^2}}…
View solution →Let f be a real valued continuous function on [0, 1] and $f(x) = x + \int\limits_0^1 {(x - t)f(t)dt}$. Then, which of the following points (x, y) lies on the curve y = f(x) ?
View solution →If $$\int\limits_{0}^{\sqrt{3}} \frac{15 x^{3}}{\sqrt{1+x^{2}+\sqrt{\left(1+x^{2}\right)^{3}}}} \mathrm{~d} x=\alpha \sqrt{2}+\beta \sqrt{3}$$, where $\alpha, \beta$ are integers,…
View solution →Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f^{\prime}(x)+f(x)=\int_\limits{0}^{2} f(t) d t$. If $f(0)=e^{-2}$, then $2 f(0)-f(2)$ is equal…
View solution →Let f : R $\to$ R be a continuous function. Then $$\mathop {\lim }\limits_{x \to {\pi \over 4}} {{{\pi \over 4}\int\limits_2^{{{\sec }^2}x} {f(x)\,dx} } \over {{x^2} - {{{\pi ^2}}…
View solution →For $0
View solution →The value of $9 \int_\limits0^9\left[\sqrt{\frac{10 x}{x+1}}\right] \mathrm{d} x$, where $[t]$ denotes the greatest integer less than or equal to $t$, is
View solution →If $$\int\limits_{{1 \over 3}}^3 {|{{\log }_e}x|dx = {m \over n}{{\log }_e}\left( {{{{n^2}} \over e}} \right)} $$, where m and n are coprime natural numbers, then ${m^2} + {n^2} -…
View solution →Let $$\int_\limits\alpha^{\log _e 4} \frac{\mathrm{d} x}{\sqrt{\mathrm{e}^x-1}}=\frac{\pi}{6}$$. Then $\mathrm{e}^\alpha$ and $\mathrm{e}^{-\alpha}$ are the roots of the equation :
View solution →The value of $${8 \over \pi }\int\limits_0^{{\pi \over 2}} {{{{{(\cos x)}^{2023}}} \over {{{(\sin x)}^{2023}} + {{(\cos x)}^{2023}}}}dx} $$ is ___________
View solution →The value of the integral $${{48} \over {{\pi ^4}}}\int\limits_0^\pi {\left( {{{3\pi {x^2}} \over 2} - {x^3}} \right){{\sin x} \over {1 + {{\cos }^2}x}}dx} $$ is equal to…
View solution →Let $[t]$ denote the greatest integer $\leq t$. Then $$\frac{2}{\pi} \int_\limits{\pi / 6}^{5 \pi / 6}(8[\operatorname{cosec} x]-5[\cot x]) d x$$ is equal to __________.
View solution →If ${I_n} = \int\limits_{{\pi \over 4}}^{{\pi \over 2}} {{{\cot }^n}x\,dx}$, then :
View solution →Let $f(x)$ be a function satisfying $f(x)+f(\pi-x)=\pi^{2}, \forall x \in \mathbb{R}$. Then $\int_\limits{0}^{\pi} f(x) \sin x d x$ is equal to :
View solution →Let f be a differentiable function satisfying $$f(x)=\frac{2}{\sqrt{3}} \int\limits_{0}^{\sqrt{3}} f\left(\frac{\lambda^{2} x}{3}\right) \mathrm{d} \lambda, x0$$ and…
View solution →The value of the integral $\int\limits_{ - 1}^1 {{{\log }_e}(\sqrt {1 - x} + \sqrt {1 + x} )dx}$ is equal to:
View solution →The integral $\int_\limits0^{\pi / 4} \frac{136 \sin x}{3 \sin x+5 \cos x} \mathrm{~d} x$ is equal to :
View solution →Let $f:R \to R$ be a function defined by : $$f(x) = \left\{ {\matrix{ {\max \,\{ {t^3} - 3t\} \,t \le x} & ; & {x \le 2} \cr {{x^2} + 2x - 6} & ; & {2 5} \cr } } \right.$$ where…
View solution →If the shortest distance between the lines $\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}$ and $\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}$ is $\frac{38}{3 \sqrt{5}} \mathrm{k}$, and…
View solution →If $x\phi (x) = \int\limits_5^x {(3{t^2} - 2\phi '(t))dt}$, x > $-$2, and $\phi$(0) = 4, then $\phi$(2) is __________.
View solution →The value of the integral $\int\limits_0^\pi {|{{\sin }\,}2x|dx}$ is ___________.
View solution →Let $[t]$ denote the greatest integer function. If $$\int_\limits{0}^{2.4}\left[x^{2}\right] d x=\alpha+\beta \sqrt{2}+\gamma \sqrt{3}+\delta \sqrt{5}$$, then…
View solution →Let f : R $\to$ R be a continuous function satisfying f(x) + f(x + k) = n, for all x $\in$ R where k 0 and n is a positive integer. If ${I_1} = \int\limits_0^{4nk} {f(x)dx}$ and…
View solution →$$\int\limits_6^{16} {{{{{\log }_e}{x^2}} \over {{{\log }_e}{x^2} + {{\log }_e}({x^2} - 44x + 484)}}dx} $$ is equal to :
View solution →The value of the integral $\int\limits_{0}^{\frac{\pi}{2}} 60 \frac{\sin (6 x)}{\sin x} d x$ is equal to _________.
View solution →If $\int_\limits{-0.15}^{0.15}\left|100 x^{2}-1\right| d x=\frac{k}{3000}$, then $k$ is equal to ___________.
View solution →$$\mathop {\lim }\limits_{x \to 0} {{\int\limits_0^{{x^2}} {\left( {\sin \sqrt t } \right)dt} } \over {{x^3}}}$$ is equal to :
View solution →$$\int_\limits0^{\pi / 4} \frac{\cos ^2 x \sin ^2 x}{\left(\cos ^3 x+\sin ^3 x\right)^2} d x \text { is equal to }$$
View solution →$$\lim_\limits{x \rightarrow 0} \frac{48}{x^{4}} \int_\limits{0}^{x} \frac{t^{3}}{t^{6}+1} \mathrm{~d} t$$ is equal to ___________.
View solution →The integral $$\int\limits_{{\pi \over 6}}^{{\pi \over 3}} {{{\tan }^3}x.{{\sin }^2}3x\left( {2{{\sec }^2}x.{{\sin }^2}3x + 3\tan x.\sin 6x} \right)dx} $$ is equal to:
View solution →Let $I_{n}(x)=\int_{0}^{x} \frac{1}{\left(t^{2}+5\right)^{n}} d t, n=1,2,3, \ldots .$ Then :
View solution →Let $$f(x) = x + {a \over {{\pi ^2} - 4}}\sin x + {b \over {{\pi ^2} - 4}}\cos x,x \in R$$ be a function which satisfies $f(x) = x + \int\limits_0^{\pi /2} {\sin (x + y)f(y)dy}$.…
View solution →The function f(x), that satisfies the condition $f(x) = x + \int\limits_0^{\pi /2} {\sin x.\cos y\,f(y)\,dy}$, is :
View solution →If $$f(\alpha)=\int\limits_{1}^{\alpha} \frac{\log _{10} \mathrm{t}}{1+\mathrm{t}} \mathrm{dt}, \alpha0$$, then $f\left(\mathrm{e}^{3}\right)+f\left(\mathrm{e}^{-3}\right)$ is…
View solution →The value of $$\int_\limits{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{(2+3 \sin x)}{\sin x(1+\cos x)} d x$$ is equal to :
View solution →Let $$f(t) = \int\limits_0^t {{e^{{x^3}}}\left( {{{{x^8}} \over {{{({x^6} + 2{x^3} + 2)}^2}}}} \right)dx} $$. If $f(1) + f'(1) = \alpha e - {1 \over 6}$, then the value of…
View solution →The value of $$\int\limits_{{{ - 1} \over {\sqrt 2 }}}^{{1 \over {\sqrt 2 }}} {{{\left( {{{\left( {{{x + 1} \over {x - 1}}} \right)}^2} + {{\left( {{{x - 1} \over {x + 1}}}…
View solution →Let g(x) = $\int_0^x {f(t)dt}$, where f is continuous function in [ 0, 3 ] such that ${1 \over 3}$ $\le$ f(t) $\le$ 1 for all t$\in$ [0, 1] and 0 $\le$ f(t) $\le$ ${1 \over 2}$…
View solution →Let $y=f(x)$ be a thrice differentiable function in $(-5,5)$. Let the tangents to the curve $y=f(x)$ at $(1, f(1))$ and $(3, f(3))$ make angles $\pi / 6$ and $\pi / 4$,…
View solution →Let $f$ be a continuous function satisfying $$\int_\limits{0}^{t^{2}}\left(f(x)+x^{2}\right) d x=\frac{4}{3} t^{3}, \forall t 0$$. Then $f\left(\frac{\pi^{2}}{4}\right)$ is equal…
View solution →If for all real triplets (a, b, c), ƒ(x) = a + bx + cx2; then $\int\limits_0^1 {f(x)dx}$ is equal to :
View solution →Let the function $f:[0,2] \rightarrow \mathbb{R}$ be defined as $$f(x)= \begin{cases}e^{\min \left\{x^{2}, x-[x]\right\},} & x \in[0,1) \\ e^{\left[x-\log _{e} x\right]}, & x…
View solution →$$\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{1 \over {{{\left( {x - {\pi \over 2}} \right)}^2}}}\int\limits_{{x^3}}^{{{\left( {{\pi \over 2}} \right)}^3}} {\cos \left(…
View solution →If $$\int\limits_0^\pi {{{{5^{\cos x}}(1 + \cos x\cos 3x + {{\cos }^2}x + {{\cos }^3}x\cos 3x)dx} \over {1 + {5^{\cos x}}}} = {{k\pi } \over {16}}} $$, then k is equal to…
View solution →If $\int\limits_{-\pi / 2}^{\pi / 2} \frac{8 \sqrt{2} \cos x \mathrm{~d} x}{\left(1+\mathrm{e}^{\sin x}\right)\left(1+\sin ^4 x\right)}=\alpha \pi+\beta \log _{\mathrm{e}}(3+2…
View solution →Let $f:(0, \infty) \rightarrow \mathbf{R}$ and $\mathrm{F}(x)=\int\limits_0^x \mathrm{t} f(\mathrm{t}) \mathrm{dt}$. If $\mathrm{F}\left(x^2\right)=x^4+x^5$, then…
View solution →Let $f:(0, \infty) \rightarrow \mathbf{R}$ be a twice differentiable function. If for some $a\ne 0, \int\limits_0^1 f(\lambda x) \mathrm{d} \mathrm{\lambda}=a f(x), f(1)=1$ and…
View solution →If $ 24 \int\limits_0^{\frac{\pi}{4}} \bigg[\sin \left| 4x - \frac{\pi}{12} \right| + [2 \sin x] \bigg] dx = 2\pi + \alpha $, where $[\cdot]$ denotes the greatest integer…
View solution →Let [.] denote the greatest integer function. If $\int_\limits0^{e^3}\left[\frac{1}{e^{x-1}}\right] d x=\alpha-\log _e 2$, then $\alpha^3$ is equal to _________.
View solution →If $\phi(x)=\frac{1}{\sqrt{x}} \int\limits_{\frac{\pi}{4}}^x\left(4 \sqrt{2} \sin t-3 \phi^{\prime}(t)\right) d t, x0$, then $\emptyset^{\prime}\left(\frac{\pi}{4}\right)$ is…
View solution →If $\int\limits_0^1 \frac{1}{\sqrt{3+x}+\sqrt{1+x}} \mathrm{~d} x=\mathrm{a}+\mathrm{b} \sqrt{2}+\mathrm{c} \sqrt{3}$, where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are rational…
View solution →The value of the integral $\int\limits_0^{\pi / 4} \frac{x \mathrm{~d} x}{\sin ^4(2 x)+\cos ^4(2 x)}$ equals :
View solution →The value of $\int\limits_0^1\left(2 x^3-3 x^2-x+1\right)^{\frac{1}{3}} \mathrm{~d} x$ is equal to :
View solution →Let for $f(x)=7 \tan ^8 x+7 \tan ^6 x-3 \tan ^4 x-3 \tan ^2 x, \quad \mathrm{I}_1=\int_0^{\pi / 4} f(x) \mathrm{d} x$ and $\mathrm{I}_2=\int_0^{\pi / 4} x f(x) \mathrm{d} x$. Then…
View solution →The value of $\int_{e^2}^{e^4} \frac{1}{x}\left(\frac{e^{\left(\left(\log _e x\right)^2+1\right)^{-1}}}{e^{\left(\left(\log _e x\right)^2+1\right)^{-1}}+e^{\left(\left(6-\log _e…
View solution →If $\mathrm{I}=\int_0^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x} \mathrm{~d} x$, then $\int_0^{2I} \frac{x \sin x \cos x}{\sin ^4…
View solution →If $I(m, n)=\int_0^1 x^{m-1}(1-x)^{n-1} d x, m, n0$, then $I(9,14)+I(10,13)$ is
View solution →If $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{96 x^2 \cos ^2 x}{\left(1+e^x\right)} \mathrm{d} x=\pi\left(\alpha \pi^2+\beta\right), \alpha, \beta \in \mathbb{Z}$, then…
View solution →Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\int\limits_0^x t f(t) d t$. If $g\left(x^3\right)=x^6+x^7$, then value of…
View solution →Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a twice differentiable function such that $f(2)=1$. If $\mathrm{F}(\mathrm{x})=\mathrm{x} f(\mathrm{x})$ for all $\mathrm{x}…
View solution →The integral $80 \int\limits_0^{\frac{\pi}{4}}\left(\frac{\sin \theta+\cos \theta}{9+16 \sin 2 \theta}\right) d \theta$ is equal to :
View solution →$4 \int_0^1\left(\frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}}\right) d x-3 \log _e(\sqrt{3})$ is equal to :
View solution →Let $(a, b)$ be the point of intersection of the curve $x^2=2 y$ and the straight line $y-2 x-6=0$ in the second quadrant. Then the integral…
View solution →Let $f:[1, \infty) \rightarrow[2, \infty)$ be a differentiable function. If $10 \int_1^x f(\mathrm{t}) \mathrm{dt}=5 x f(x)-x^5-9$ for all $x \geqslant 1$, then the value of…
View solution →Let the domain of the function $f(x)=\log _2 \log _4 \log _6\left(3+4 x-x^2\right)$ be $(a, b)$. If $\int_0^{b-a}\left[x^2\right] d x=p-\sqrt{q}-\sqrt{r}, p, q, r \in \mathbb{N},…
View solution →The value of $\int_\limits{-1}^1 \frac{(1+\sqrt{|x|-x}) e^x+(\sqrt{|x|-x}) e^{-x}}{e^x+e^{-x}} d x$ is equal to
View solution →Let $f(x)+2 f\left(\frac{1}{x}\right)=x^2+5$ and $2 g(x)-3 g\left(\frac{1}{2}\right)=x, x0$. If $\alpha=\int_1^2 f(x) \mathrm{d} x$, and $\beta=\int_1^2 g(x) \mathrm{d} x$, then…
View solution →Let f(x) be a positive function and $I_{1} = \int\limits_{-\frac{1}{2}}^{1} 2x \, f(2x(1-2x)) \, dx$ and $I_{2} = \int\limits_{-1}^{2} f(x(1-x)) \, dx$. Then the value of…
View solution →The integral $\int\limits_{-1}^{\frac{3}{2}} \left(| \pi^2 x \sin(\pi x) \right|) dx$ is equal to:
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