If the value of the integral ∫ (x^2cosx / 1+π^x + 1+sin^2x / 1+e^sinx^2023)dx; x ∈ [-π/2, π/2] = π/4 (π + a) − 2

Question

If the value of the integral \(\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^{2} \cos x}{1+\pi^{x}}+\frac{1+\sin ^{2} x}{1+e^{\sin x^{2023}}}\right) d x=\frac{\pi}{4}(\pi+a)-2\) then the value of \(a\) is

(1) \(3\)

(2) \(-\frac{3}{2}\)

(3) \(2\)

(4) \(\frac{3}{2}\)

Answer 1

Correct option is (1) \(3\)

\(I=\int\limits_{-\pi / 2}^{\pi / 2}\left(\frac{x^{2} \cos x}{1+\pi^{x}}+\frac{1+\sin ^{2} x}{1+e^{\sin x^{2023}}}\right) d x\)

\(I=\int\limits_{-\pi / 2}^{\pi / 2}\left(\frac{x^{2} \cos x}{1+\pi^{-x}}+\frac{1+\sin ^{2} x}{1+e^{\sin (-x)^{2023}}}\right) d x\)

On Adding, we get

\(2 I=\int\limits_{-\pi / 2}^{\pi / 2}\left(x^{2} \cos x+1+\sin ^{2} x\right) d x\)

On solving

\(\mathrm{I}=\frac{\pi^{2}}{4}+\frac{3 \pi}{4}-2\)

\(\mathrm{a}=3\)