Question

Calcule a integral [(cosec² x)/(secx)] .dx

Answer 1

Temos:

$\displaystyle \mathsf{\int \frac{cossec^2 \, x}{sec \, x} \, dx}$

E de acordo com as identidades trigonométricas:

$\displaystyle \mathsf{cossec^2 \, x = \frac{1}{sen^2 \, x}} \\ \\ \\ \mathsf{sec \, x = \frac{1}{cos \, x}}$

Ficamos com:

$\displaystyle \mathsf{\int \frac{cos \, x}{sen^2 \, x} \, dx} \\ \\ \\ \mathsf{\int \frac{1}{sen^2 \, x} \cdot cos \, x \, dx} \\ \\ \\ \mathsf{u = sen \, x} \\ \\ \mathsf{du = cos \, x \, dx} \\ \\ \\ \mathsf{\int \frac{1}{u^2} \, du} \\ \\ \\ \mathsf{-\frac{1}{u}+c} \\ \\ \\ \mathsf{-\frac{1}{sen \, x} + c} \\ \\ \\ \boxed{\boxed{ \mathsf{-cossec \, x + c} }}$