Thread: Integral- trig substitution?

1. Integral- trig substitution?

Hi. I'm trying to solve this integral. I think I may need a trig substitution, but I've tried a ton and I'm lost. Any specific help would be much appreciated.

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

The solution is

$\frac{x}{z^2(x^2+z^2)^{1/2}}$

2. Originally Posted by daberkow
Hi. I'm trying to solve this integral. I think I may need a trig substitution, but I've tried a ton and I'm lost. Any specific help would be much appreciated.

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

The solution is

$\frac{x}{z^2(x^2+z^2)^{1/2}}$
Assuming $z$ is a constant:
Yes. Triginometric Substitution. Let $x=ztan(\theta)$ then $dx=zsec^2(\theta)d\theta$.
Post your work; to see why did you stuck.

3. $
\int\frac{1}{(x^2+z^2)^\frac{3}{2}}dx
$

$\int\frac{zsec^2\theta}{(z^2+z^2tan^2\theta)^\frac {3}{2}}d\theta$

$\int\frac{zsec^2\theta}{((z^2(1+tan^2\theta))^\fra c{3}{2}}d\theta$

$\int\frac{zsec^2\theta}{(z^2(sec^2\theta))^\frac{3 }{2}}d\theta$

$\int\frac{sec^2\theta}{z^2(sec^2\theta))^\frac{3}{ 2}}d\theta$

$\int\frac{1}{z^2(sec^2\theta)^\frac{1}{2}}d\theta$

$\int\frac{1}{z^2(sec\theta)}d\theta$

I went wrong somewhere...

4. Originally Posted by daberkow
$
\int\frac{1}{(x^2+z^2)^\frac{3}{2}}dx
$

$\int\frac{zsec^2\theta}{(z^2+z^2tan^2\theta)^\frac {3}{2}}d\theta$

$\int\frac{zsec^2\theta}{((z^2(1+tan^2\theta))^\fra c{3}{2}}d\theta$

$\int\frac{zsec^2\theta}{(z^2(sec^2\theta))^\frac{3 }{2}}d\theta$

$\int\frac{sec^2\theta}{z^2(sec^2\theta))^\frac{3}{ 2}}d\theta$

$\int\frac{1}{z^2(sec^2\theta)^\frac{1}{2}}d\theta$

$\int\frac{1}{z^2(sec\theta)}d\theta$

I went wrong somewhere...
Try using $\frac{1}{\sec\theta}=\cos\theta$

5. Originally Posted by daberkow
I went wrong somewhere...
No, keep going...

http://www.mathhelpforum.com/math-he...tml#post457096

6. Originally Posted by daberkow
$
\int\frac{1}{(x^2+z^2)^\frac{3}{2}}dx
$

$\int\frac{zsec^2\theta}{(z^2+z^2tan^2\theta)^\frac {3}{2}}d\theta$

$\int\frac{zsec^2\theta}{((z^2(1+tan^2\theta))^\fra c{3}{2}}d\theta$

$\int\frac{zsec^2\theta}{(z^2(sec^2\theta))^\frac{3 }{2}}d\theta$

$\int\frac{sec^2\theta}{z^2(sec^2\theta))^\frac{3}{ 2}}d\theta$

$\int\frac{1}{z^2(sec^2\theta)^\frac{1}{2}}d\theta$

$\int\frac{1}{z^2(sec\theta)}d\theta$

I went wrong somewhere...
Good. No mistakes.
You almost done.
$\int\frac{1}{z^2(sec\theta)}d\theta=\frac{1}{z^2}\ int cos(\theta) d\theta$