# Integral Question

• Jun 26th 2007, 07:05 PM
cinder
Integral Question
Need some help..

$\int \frac{\sin(x)}{1+ \cos^2(x)}dx$

Where do I begin? I imagine I need some identities and substitution, but I'm not sure.
• Jun 26th 2007, 07:08 PM
topsquark
Quote:

Originally Posted by cinder
Need some help..

$\int \frac{sin(x)~dx}{1+ cos^2(x)}$

Substitution.

Let $y = cos(x)$. Then $dy = -sin(x) dx$. So your integral becomes:
$\int \frac{sin(x)~dx}{1+ cos^2(x)} = \int \frac{-dy}{1 + y^2}$

Can you do the integral now?

-Dan
• Jun 26th 2007, 07:15 PM
cinder
Quote:

Originally Posted by topsquark
Substitution.

Let $y = cos(x)$. Then $dy = -sin(x) dx$. So your integral becomes:
$\int \frac{sin(x)~dx}{1+ cos^2(x)} = \int \frac{-dy}{1 + y^2}$

Can you do the integral now?

-Dan

Okay, I see now. Let me try to integrate it.
• Jun 26th 2007, 07:23 PM
cinder
Is it $-(\arctan(u)+C)$?
• Jun 26th 2007, 07:25 PM
topsquark
Quote:

Originally Posted by cinder
Is it $-(\arctan(u)+C)$?

Yup!

-Dan
• Jun 26th 2007, 07:26 PM
cinder
Quote:

Originally Posted by topsquark
Yup!

-Dan

Whoops... I'd throw $\cos(x)$ in place of $u$ though, right?
• Jun 26th 2007, 07:28 PM
topsquark
Quote:

Originally Posted by cinder
Whoops... I'd throw $\cos(x)$ in place of $u$ though, right?

Yup! Yup! ;)

Actually, if you know the quadrant x is in you can simplify the $atn(cos(x))$ expression somewhat. But if it's an indefinite integral then you'll have to leave it like that.

-Dan