Integration by reduction question

Hi I have seen on Wikipedia that the formula for integration of x^n.(ax+b)^1/2 by reduction is given at http://en.m.wikipedia.org/wiki/Integration_by_reduction_formulae but I do not understand how they come across this. If someone could explain I would be really grateful

Hi I have seen on Wikipedia that the formula for integration of x^n.(ax+b)^1/2 by reduction is given at http://en.m.wikipedia.org/wiki/Integration_by_reduction_formulae but I do not understand how they come across this. If someone could explain I would be really grateful

$I_n=\displaystyle \int x^n\sqrt{ax+b} \ dx$
If we then integrate this by parts as is normal with reduction formulae with: $u=x^n$, $v=\sqrt{ax+b}$
We get:


The step which is, I imagine, tripping you up is this one:


And finally, doing some simple rearranging yields the given result.

Please do try to follow through the algebra and work the details out for yourself

Hi. Thanks. Do I had actually got to the part where I alter (ax+b)^3/2 to just (ax+b). (ax+b)^1/2. Do I have to do by parts again. I can't seem to understand how you've split that up tbh