Just another one of those nights...

[Daniel_Feldman]

I'm drawing blanks. Help would be awesome.

The problem:

Integrate (-5x)/(x^2+5)^(3/2) dx


Here's what I've worked out so far:

Rewriting the denominator as (root(x^2+5))^3, I then let x=tan(theta), so dx=sec^2(theta) dtheta.

So I rewrote the integral as (-5tantheta)/(root(tan^2(theta)+5))^3 * sec^2(theta) dtheta.

Here is where I'm staring at it. Have I gone wrong already, or do I simply not see where to proceed?

 

[Unco]

The standard substitution is: \(\displaystyle \L x \, = \, \sqrt{5}\tan{\theta}\)

 

[Daniel_Feldman]

Thanks Unco. With the proper substitution, it suddenly works. How interesting .

 

[soroban]

Hello, Dan!

It's easier than you think . . .

\(\displaystyle \L\int \frac{-5x}{(x^2\,+\,5)^{\frac{3}{2}}}\,dx\)

Let \(\displaystyle u\:=\:x^2\,+\,5\)