Question

The value of ${\int }_{-\pi /2}^{\pi /2}\frac{dx}{\sin {}^{3}x+\sin x}$is

• A. $0$
• B. $2$
• C. $1$
• D. none of these

Answer 1

The correct option is A $0$

Let $I={\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{dx}{{\sin }^{3}x+\sin x}$
Using ${\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f(a+b-x)dx$
$I={\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{dx}{{\sin }^3(-x)+\sin (-x)}dx$
$=-{\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{dx}{{\sin }^{3}x+\sin x}dx=-I\therefore 2I=I\Rightarrow I=0$