Question

Integral de sec ^ 2 x / cosec x dx

Answer 1

$\int_{}^{} {sec^2(x) \over cosec(x)} \, dx \\\\ \int_{}^{} {1 \over cos^2x } \times sen(x) \, dx \\\\ \int_{}^{} {sen(x) \over cos^2(x) } \, dx \\\\ \int_{}^{} tan(x)sec(x) \, dx$

Substituindo o diferencial dx = 1/u' du, com
u = sec(x) e u' = tan(x)sec(x), temos:

$\int_{}^{} tan(x)sec(x) \times { 1 \over tan(x)sec(x) } \, du \\\\ \int_{}^{} 1 \, du = u$

Devolva o valor:

$u = sec(x)$

Resultado:

$\int_{}^{} {sec^2(x) \over cosec(x)} \, dx = \boxed{ sec(x) + C{,} \: C \in \mathbb{R}}$