Integral of (x^2)/(x^2-9)^(3/2) and use trigonometric substitution

Question

Integral of (x^2)/(x^2-9)^(3/2) use trigonometric substitution

 

I simplified it to

 

integral of ((sec of theta)^3)/(tan of theta)^2, but I couldn't figure out how to correctly find the integral from that point.

 

Please help!

Answer 1

Do you know integration by parts? UV - int VdU?

 

Split your answer into secø   *    (secø)^2 / (tanø)^2 dø

so you get secø     *    (cscø)^2 dø.  

 

Does this help you finish?

Answer 2

I believe it is sec²θ /3tan³θ

 

so you can call u = tanθ

du = sec²θ dθ

 

and the integral becomes   du/3u³