I found some integral problems online and have been using them to practice. I'm trying to integrate x(x + 1)^-5 with intervals of infinity and zero.

The solution here given seems to take advantage of a shortcut. That integral is changed to the integral of y^-4 dy minus the integral of y^-5 dy, both with boundaries of infinity and 1, instead of infinity and zero.

When I check the integral at Wolfram Alpha I get the integral as being (4x + 1) / 12 (x + 1)^4

Which ties out to the same answer using the shortcut of the y^-4 minus y^-5 with the changed interval.

Can someone walk through integrating x(x + 1)^-5 to get to the (4x+1) / 12(x+1)^4,

and then also explain why the shortcut works of taking y^-4 - y^-5?

I've attached the solution file, which shows the shortcut.

Have a nice day!

Chris