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U Substitution with definite integral help

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:

Attachment 25996

Thanks guys! :)

Re: U Substitution with definite integral help

I would let:

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

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

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

Re: U Substitution with definite integral help

Quote:

Originally Posted by

**MarkFL2** I would let:

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

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

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

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

Re: U Substitution with definite integral help

Equivalently, $\displaystyle \frac{x- 1}{\sqrt{x}}= \frac{x}{x^{1/2}}- \frac{1}{x^{1/2}}= x^{1- 1/2}- x^{0- 1/2}= x^{1/2}- x^{-1/2}$

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