Tricky u-Substitution

by Donny
(Atlanta GA)

I have been trying to solve the indefinite integral of (x+3)(x-1)^1/2 dx. I can't figure out how to get rid of x+3 using u=(x-1)^1/2 or u=x-1. Am I picking the wrong u substitution? Any help is appreciated!

Answer by Pablo:

This integral is quite interesting, I've never seen something similar given as an exercise before. However, it is solvable by simple substitution.

The easiest choice is taking:

If we sum 4 to both sides of the definition of u:

So, there we have everything we need to make the substitution. Now:

We distribute the product inside and then integrate directly:

Now, substituting back:

Taking the derivative you can verify that this answer is indeed correct.

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