If the value of the integral $$\int_\limits{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^2 \cos x}{1+\pi^x}+\frac{1+\sin ^2 x}{1+e^{\sin x^{2123}}}\right) d x=\frac{\pi}{4}(\pi+a)-2$$, then the value of $a$ is
Solution
<p>$$\begin{aligned}
& I=\int_\limits{-\pi / 2}^{\pi / 2}\left(\frac{x^2 \cos x}{1+\pi^x}+\frac{1+\sin ^2 x}{1+e^{\sin x^{2023}}}\right) d x \\
& I=\int_\limits{-\pi / 2}^{\pi / 2}\left(\frac{x^2 \cos x}{1+\pi^{-x}}+\frac{1+\sin ^2 x}{1+e^{\sin (-x)^{2023}}}\right) d x
\end{aligned}$$</p>
<p>On Adding, we get</p>
<p>$2 I=\int_\limits{-\pi / 2}^{\pi / 2}\left(x^2 \cos x+1+\sin ^2 x\right) d x$</p>
<p>On solving</p>
<p>$$\begin{aligned}
& \mathrm{I}=\frac{\pi^2}{4}+\frac{3 \pi}{4}-2 \\
& \mathrm{a}=3
\end{aligned}$$</p>
About this question
Subject: Mathematics · Chapter: Definite Integration · Topic: Properties of Definite Integrals
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