U Substitution with definite integral help

**billb91** · November 30th 2012, 12:39 PM

Hi all,

I have a homework problem (not graded, so I'm not coming here to boost my grade if that was anybody's concern :P) that I got caught up on. I'm not sure what to set u equal to, I originally though square root of x but that was getting extremely messy. I was hoping you guys could help walk me through this one to give me an idea of where I should go with it.

Here's the problem:

U Substitution with definite integral help-mathproblem.png

Thanks guys!

**MarkFL** · November 30th 2012, 12:46 PM

I would let:

$u=\sqrt{x}$ and since $1\le x\le4$ we may state:

$u^2=x\,\therefore\,2u\,du=dx$ and we have:

$2\int_1^2 u^2-1\,du$

**billb91** · November 30th 2012, 01:21 PM

Originally Posted by MarkFL2

I would let:

$u=\sqrt{x}$ and since $1\le x\le4$ we may state:

$u^2=x\,\therefore\,2u\,du=dx$ and we have:

$2\int_1^2 u^2-1\,du$

Thanks so much! It didn't even cross my mind to set it equal to x.

**HallsofIvy** · November 30th 2012, 01:42 PM

Equivalently, $\frac{x- 1}{\sqrt{x}}= \frac{x}{x^{1/2}}- \frac{1}{x^{1/2}}= x^{1- 1/2}- x^{0- 1/2}= x^{1/2}- x^{-1/2}$
so the integral become $\int_1^4 x^{1/2}- x^{-1/2}dx$ and you can integrate both parts by using $\int x^ndx= \frac{x^{n+1}}{n+1}+ C$ .

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