JEE Main 2022 · 25 Jun · Shift 1 · Mathematics Q2

MathematicsSingle correct+4 / -1Question 2 of 90

The value of $\displaystyle\int_0^\pi\frac{e^{\cos x}\sin x}{\left(1+\cos^2x\right)\left(e^{\cos x}+e^{-\cos x}\right)}dx$ is equal to :

\dfrac{\pi^2}4

\dfrac{\pi^2}2

\dfrac\pi4

\dfrac\pi2

Answer and worked solution

Official answer: C

Substitute $t=\cos x$ . $I=\displaystyle\int_{-1}^1\frac{e^t}{(1+t^2)(e^t+e^{-t})}dt$ .

Use $t\to-t$ . The same integral with $e^{-t}$ on top; adding the two forms gives $2I=\displaystyle\int_{-1}^1\frac{dt}{1+t^2}=\frac\pi2$ .

So $I=\dfrac\pi4$ .

(C)

Topics:Properties of definite integrals Substitution

JEE Main 2022 · 25 Jun · Shift 1, Mathematics question 2