# Math Help - Integral Question

1. ## 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.

2. 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

3. 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.

4. Is it $-(\arctan(u)+C)$?

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

-Dan

6. Originally Posted by topsquark
Yup!

-Dan
Whoops... I'd throw $\cos(x)$ in place of $u$ though, right?

7. 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