# How do you integrate y=sinx/tanx using the quotient rule?

Nov 13, 2016

$\int \sin \frac{x}{\tan} x \mathrm{dx} = \sin x + C$

#### Explanation:

The quotient rule is a formula for finding the derivative of $\frac{u}{v}$, rather than the integral of $\frac{u}{v}$. There is no formula for finding integral of $\frac{u}{v}$

The integral you have specified can be deal with as follows;

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

$\therefore \int \sin \frac{x}{\tan} x \mathrm{dx} = \int \sin x \cdot \left(\cos \frac{x}{\sin} x\right) \mathrm{dx}$

$\therefore \int \sin \frac{x}{\tan} x \mathrm{dx} = \int \cos x \mathrm{dx}$

$\therefore \int \sin \frac{x}{\tan} x \mathrm{dx} = \sin x + C$