"The function f(x), that satisfies the condition $f(x) = x + \int\limits_0^{\pi /2} {\sin x.\cos y\,f(y)\,dy}$, is :"

Question

Accepted Answer

"$f(x) = x + \int\limits_0^{\pi /2} {\sin x\cos y\,f(y)\,dy}$

$f(x) = x + sinx\underbrace {\int_0^{\pi /2} {\cos y\,f(y)\,dy} }_K$

$\Rightarrow f(x) = x + K\sin x$

$\Rightarrow f(y) = y + K\sin y$

Now, $$K = \int_0^{\pi /2} {\mathop {y\cos y\,dy}\limits_{Apply\,IBP} } + K\int_0^{\pi /2} {\mathop {\cos y\sin y\,dy}\limits_{Put\,\sin y = t} } $$

$$K = \left( {y\sin y} \right)_0^{\pi /2} - \int_0^{\pi /2} {\sin y dy + K\int_0^1 {t\,dt} } $$

$\Rightarrow K = {\pi \over 2} - 1 + K\left( {{1 \over 2}} \right)$

$\Rightarrow K = \pi - 2$

So, $f(x) = x + (\pi - 2)\sin x$

Option (d)"

The function f(x), that satisfies the condition
$f(x) = x + \int\limits_0^{\pi /2} {\sin x.\cos y\,f(y)\,dy}$, is :

Solution

About this question

Drill 25 more like these. Every day. Free.

The function f(x), that satisfies the condition $f(x) = x + \int\limits_0^{\pi /2} {\sin x.\cos y\,f(y)\,dy}$, is :

Solution

About this question

More Definite Integration questions

Drill 25 more like these. Every day. Free.

The function f(x), that satisfies the condition
$f(x) = x + \int\limits_0^{\pi /2} {\sin x.\cos y\,f(y)\,dy}$, is :