# How can I find the integral of cosx/sinx+cosx?

Jan 22, 2018

$\int \cos \frac{x}{\sin} x + \cos x \mathrm{dx} = \ln \left\mid \sin \right\mid x + \sin x + C$

#### Explanation:

Given: $\int \cos \frac{x}{\sin} x + \cos x \mathrm{dx}$

We are essentially finding two integrals so we really have

$\int \cos \frac{x}{\sin} x \mathrm{dx} + \int \cos \mathrm{dx}$

We'll solve each integral separately

$\int \cos \frac{x}{\sin} x \mathrm{dx}$

Make a substitution

Let $u = \sin x \implies \mathrm{du} = \cos x \mathrm{dx}$

$\int \frac{1}{u} \mathrm{du}$

Use: color(blue)(int1/udu=lnabsu

$= \ln \left\mid u \right\mid$

Reverse the substitution:

$= \ln \left\mid \sin \right\mid x$

For $\int \cos x \mathrm{dx}$

This is a common integral whose antiderivative is $\sin x$

Putting it all together

$\int \cos \frac{x}{\sin} x + \cos x \mathrm{dx} = \ln \left\mid \sin \right\mid x + \sin x + C$