# Trig Substitution

• Nov 16th 2009, 02:32 PM
JJ007
Trig Substitution
$\int \frac {1}{(x^2+a^2)^{3/2}}dx$

$\int \frac {1}{(\sqrt{(x^2+a^2)})^3}dx$

$x=a tan\theta$
$dx=sec^2\theta$

$\sqrt{(x^2+a^2)}= \sqrt{a^2tan^2\theta+a^2}= asec\theta\ ?$

$\int \frac {1}{({asec\theta})^3}sec^2\theta\ ?$
• Nov 16th 2009, 05:18 PM
Scott H
Looks good to me.

We must remember that $\sqrt{\sec^2\theta}=|\sec\theta|$, but since $\theta=\arctan\frac{x}{a}$, $\sec\theta$ is always positive.
• Nov 17th 2009, 05:59 AM
JJ007
Quote:

Originally Posted by Scott H
Looks good to me.

We must remember that $\sqrt{\sec^2\theta}=|\sec\theta|$, but since $\theta=\arctan\frac{x}{a}$, $\sec\theta$ is always positive.

Ok thanks, I'm not sure what to do next though. u substitution? trig identities?

$\int \frac {1}{({asec\theta})^3}sec^2\theta=\$
• Nov 17th 2009, 06:19 AM
11rdc11
Quote:

Originally Posted by JJ007
Ok thanks, I'm not sure what to do next though. u substitution? trig identities?

$\int \frac {1}{({asec\theta})^3}sec^2\theta=\$

$\frac{1}{a^3} \int \cos{\theta}d\theta$
• Nov 17th 2009, 06:31 AM
JJ007
Quote:

Originally Posted by 11rdc11
$\frac{1}{a^3} \int \cos{\theta}d\theta$

$\frac{sin{\theta\ d\theta } }{a^3}+c$
So is this the full answer to the problem or does it involve a triangle at some point?
• Nov 17th 2009, 07:34 AM
11rdc11
Quote:

Originally Posted by JJ007
$\frac{sin{\theta\ d\theta } }{a^3}+c$
So is this the full answer to the problem or does it involve a triangle at some point?

One more step, back sub for your the trig sub

$\frac{x}{a^3\sqrt{a^2+x^2}}+C$