Browse Questions

# Evaluate:$\int_{-\pi}^\pi (\sin ^{-1}x+x^{295})$

Toolbox:
• (i) $\int \limits_a^b f(x)dx=F(b)-F(a)$
• (ii) If $f(-x) =-f(x),$ then the function is an odd function. hence for an odd function $\int \limits_{-a}^af(x)dx=0$
Given $I=\int_{-\pi}^\pi (\sin ^{-1}x+x^{295})_{dx}^{-a}$

Consider $\sin ^{-1}(x)+x^{295}$ Replace x by -x

$\sin ^{-1}(-x)+(-x)^{295}$

This is equal to 0,hence it is an odd function

We know $\int \limits_{-a}^af(x)dx=0$ if f(x) is an odd function

Therefore $I=\int_{-\pi}^\pi (\sin ^{-1}(x)+x^{295}){dx}=0$