Integration Problem

kelxx

Can anyone solve this?

Evaluate the following using substitution:

∫ (x^3)(x^2 + 1)^(-1/2)

Thanks!!

Eliot Rosewater

Do you've any ideas so far?

kelxx

I dont realy know where to start. Like what would i let u equal to?

Eliot Rosewater

In fairness, there's a pretty small number of things you could let it equal to. Try a few of them, and say where you get stuck.

kelxx

If you let

u = (x^2 + 1)
du = 2x dx
du/2 = x dx

but what do u do with the x^3?

or if you let u = x^3 it doesnt work.
Really stuck!!

Eliot Rosewater

You're on the right track. Using the fact that du/2=xdx only gets rid of one of the power of x. I.e.,

[LATEX]\displaystyle \int \frac{x^3}{\sqrt{x^2+1}}dx[/LATEX]

becomes

[LATEX]\displaystyle \int \frac{x^2}{\sqrt{u}}\frac{du}{2}[/LATEX]

So you need to write x squared in terms of u to get rid of it?

kelxx

So

u = x^2 + 1
X^2 = u - 1

then
1/2 ∫(u-1)/sqrt(u) du

Then by multiplying out get:

1/2∫ (u^1/2) - u^(-1/2)

Would that be correct so far?

Eliot Rosewater

Yup.

With all of these integrals you should just shoot for a final answer; you can then differentiate it to make sure it's correct.

kelxx

Thanks a million for the help!!!

MathsArthur

the simplest substitution is u=(1+x^2)^(1/2) then

dx = du * (1+x^2)^(1/2)/x and this gives

INT (u^2-1)du which solves to give

y=(1+x^2)^(3/2)/3 - (1+x^2)^(1/2) as the solution.