Question

Evaluate $\int se{c}^{2}x.cose{c}^{2}xdx$ on $I\subset R\left(\right\{n\pi ;n\in Z\}\cup \left\{\right(2n+1)\frac{\pi }{2}:n\in Z\})$

Answer 1

$\int se{c}^{2}xcose{c}^{2}xdx$
$=\int se{c}^{2}x(1+co{t}^{2}x)dx$
$=\int se{c}^{2}x+se{c}^{2}x.co{t}^{2}xdx$
$=\int se{c}^{2}x.dx+\int \frac{1}{co{s}^{2}x}.\frac{co{s}^{2}x}{si{n}^{2}x}dx$
$=tanx+\int cose{c}^{2}xdx$
$=tanx-cotx+c$