Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{4^x}{4^x+2}$ and $$M=\int_\limits{f(a)}^{f(1-a)} x \sin ^4(x(1-x)) d x, N=\int_\limits{f(a)}^{f(1-a)} \sin ^4(x(1-x)) d x ; a \neq \frac{1}{2}$$. If $\alpha M=\beta N, \alpha, \beta \in \mathbb{N}$, then the least value of $\alpha^2+\beta^2$ is equal to __________.

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…

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 :

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…

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…

$$\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 /…