Expert: Steve Holleran - 9/29/2007

Question
hello i was wondering what the answer for the integral of
(t^3((1-t^2)^(-1/2)))  i think you have to use integration by parts and substituation im not sure cheers

Answer
Hi Mark,

Okay, I think we've got this one down.

The substitution you want is u = 1-t^2.

Then t =(1-u)^1/2  and dt = 1/2 * (1-u)^-1/2 * -1 * du


And th integral becomes

INT[ (1-u)^3/2 * -1/2 * 1/(1-u)^1/2 * du / u^1/2]

= -1/2 * INT [(1-u) / u^1/2 * du]

= -1/2 * INT [u^-1/2 - u^1/2  * du]

= -u^1/2 + 1/2 * 2/3 * u^3/2 + C

= -sqrt(1-t^2) + 1/3 * (1-t^2)^3/2 + C

I checked it out by differentiating, and thankfully, it works out.

Good luck Mark.

Steve