Question

Evaluate: ${\int }_{-\pi /4}^{\pi /4}\frac{1}{1-\sin x}dx$

Answer 1

$\Rightarrow I={\int }_{-\pi /4}^{\pi /4}\frac{1}{1+sinx}dx$ By of properties of integral, ${\int }_a^{b}f\left(n\right)dx={\int }_a^{b}f(a+b-x)dx$

$\Rightarrow I={\int }_{-\pi 4}^{\pi 4}\frac{1}{1+sinx}dx\Rightarrow 2I={\int }_{-\pi 4}^{\pi 4}\frac{2}{1-si{n}^{2}x}dx$

$\Rightarrow I=\int se{c}^{2}xdx=tanx{\int }_{-\pi 4}^{\pi 4}=1-(-1)=2$.