If the value of the integral _-( )/(2)^( )/(2)(x^2 x1+ ^x+1+ ^2x1+e^ x^2123)dx=( )/(4)( +a…

Question

If the value of the integral $\int _{-\frac{\pi }{2}}^{\frac{\pi }{2}}(\frac{{x}^{2}\cos x}{1+{\pi }^{x}}+\frac{1+{\sin }^{2}x}{1+{e}^{\sin {x}^{2123}}})dx=\frac{\pi }{4}(\pi +a)-2$, then the value of $a$ is

Accepted Answer

$\begin{gathered} I=\int _{-\pi /2}^{\pi /2}(\frac{{x}^{2}\cos x}{1+{\pi }^{x}}+\frac{1+{\sin }^{2}x}{1+{e}^{\sin {x}^{2023}}})dx \ I=\int _{-\pi /2}^{\pi /2}(\frac{{x}^{2}\cos x}{1+{\pi }^{-x}}+\frac{1+{\sin }^{2}x}{1+{e}^{\sin (-x{)}^{2023}}})dx \end{gathered}$

On Adding, we get

$2I=\int _{-\pi /2}^{\pi /2}({x}^{2}\cos x+1+{\sin }^{2}x)dx$

On solving

$\begin{gathered} I=\frac{{\pi }^{2}}{4}+\frac{3\pi }{4}-2 \ a=3 \end{gathered}$

Solutions · 1