Integral Question

**cinder** · June 26th 2007, 06:05 PM

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.

**topsquark** · June 26th 2007, 06:08 PM

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

**cinder** · June 26th 2007, 06:15 PM

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.

**cinder** · June 26th 2007, 06:23 PM

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

**topsquark** · June 26th 2007, 06:25 PM

Originally Posted by cinder

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

Yup!

-Dan

**cinder** · June 26th 2007, 06:26 PM

Originally Posted by topsquark

Yup!

-Dan

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

**topsquark** · June 26th 2007, 06:28 PM

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

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