QJEEM21Mac0cb: JEE Main 2021 Mathematics - Class 12 JEEM21Mac0cb Solution: The function f(x), that satisfies the condition , is :

Question

Verified Solution • 2025–26 Session moderate

The function f(x), that satisfies the condition

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

, is :

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)

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