Question

 

 

Evaluate ${\displaystyle {\int }_{-2}^2\frac{x^2}{1+5^x}dx}$

 

 

Answer 1

 

 Consider the given integral

${\displaystyle I={\int }_{-2}^2\frac{x^2}{1+5^x}dx}$

Let us use the property,

${\displaystyle {\int }_a^{b}f\left(x\right)dx={\int }_b^{a}f(a+b-x)dx}$

${\displaystyle \therefore I={\int }_{-2}^2\frac{{(-x)}^2}{1+5^{-x}}dx}$

 ${\displaystyle ={\int }_{-2}^2\frac{5^x{x}^2}{1+5^x}dx}$

 Adding equations (1) and (2), we have,

${\displaystyle 2I={\int }_{-2}^2\frac{1+5^x}{1+5^x}\times x^{2}dx}$

${\displaystyle ={\int }_{-2}^2{x}^{2}dx}$

${\displaystyle ={\left[\frac{x^3}{3}\right]}^2}$

${\displaystyle =\frac{1}{3}[8-\left(8\right)]}$

${\displaystyle =\frac{1}{3}\left[16\right]}$

${\displaystyle \Rightarrow I=\frac{8}{3}}$