# What is int(sinx)/(cos^2 x) dx?

Apr 21, 2018

$\sec x + C$

#### Explanation:

$\int \sin \frac{x}{{\cos}^{2} \left(x\right)} \mathrm{dx}$

using u-substitution:
let $u = \cos x$. then $\mathrm{du} = - \sin x \mathrm{dx}$ or $- \mathrm{du} = \sin x \mathrm{dx}$

substituting into the original integral:
$\int - \frac{1}{u} ^ 2 \mathrm{du}$

integrate that with power rule:

$\frac{1}{u} + C$

substitute $u = \cos x$:

$\frac{1}{\cos} x + C = \sec x + C$

Apr 21, 2018

$\sec x + C$

#### Explanation:

Based on the "inverse" nature of derivatives and antiderivatives (found through integrals), if we know that $\frac{d}{\mathrm{dx}} \sec x = \sec x \tan x$, then $\int \sec x \tan x \mathrm{dx} = \sec x + C$.

Here, using $\frac{1}{\cos} x = \sec x$ and $\tan x = \sin \frac{x}{\cos} x$, we see that

$\int \sin \frac{x}{\cos} ^ 2 x \mathrm{dx} = \int \frac{1}{\cos} x \left(\sin \frac{x}{\cos} x\right) \mathrm{dx} = \int \sec x \tan x \mathrm{dx} = \sec x + C$