- #1

- 71

- 0

## Homework Statement

[tex]\int \frac{4}{x^{2}\sqrt{81-x^{2}}} dx[/tex]

## Homework Equations

## The Attempt at a Solution

Since the radical is of the form [tex]a^2-x^2[/tex], I'm using the substitution [tex]x=asin\theta[/tex].

[tex]x = 9sin\theta[/tex]

[tex]dx = 9cos\theta d\theta[/tex]

[tex]dx = 9cos\theta d\theta[/tex]

Using this x value, I solved the radical and use the trig identity to replace 1-sin^2 with cos^2.

[tex]\sqrt{81-x^{2}}[/tex]

[tex]\sqrt{81 - (9sin\theta)^{2}}[/tex]

[tex]\sqrt{81(1-sin^2\theta)}[/tex]

[tex]\sqrt{81cos^2\theta)}[/tex]

[tex]9cos\theta[/tex]

[tex]\sqrt{81 - (9sin\theta)^{2}}[/tex]

[tex]\sqrt{81(1-sin^2\theta)}[/tex]

[tex]\sqrt{81cos^2\theta)}[/tex]

[tex]9cos\theta[/tex]

Then I threw everything back into my original integral.

[tex]\int \frac{36cos\theta}{81sin^2\theta9cos\theta} d\theta[/tex]

Canceling and simplifying...

[tex]\int \frac{4cos\theta}{81sin^2\theta} d\theta[/tex]

This is where I get lost. I don't think I'm on the right track. I've watched several demonstrations of this kind of problem, and they all work out much better than this. Usually, I think, because there's a 1 on top instead of a 4. Any hints would be great.