Question

Calcule a integral indefinida:

∫ tgx (1 + (senx.secx)²) dx

Passo a passo, por favor.

Answer 1

$\int\ {\tan x(1+(\sin x\sec x)^2)} \, dx \\\\ \int\ {\tan x(1+(\tan x)^2)} \, dx\\\\ \int\ {\tan x(\sec^2x)} \, dx\\\\\int\ {\dfrac{\sin x}{\cos^3 x}} \, dx\\\\u=\cos x\\du=-\sin x\ dx\\\\dx=\dfrac{-du}{\sin x}\\\\ \int{-\dfrac{1}{u^3}}\, du \\\\-(-\dfrac{1}{2u^2})=\dfrac{1}{2\cos^2 x}=\boxed{\dfrac{\sec^2x}{2}+C}$