Thread: [SOLVED] Evaluating this indefinite integral

The indefinite integral of x^3sq root of x^2+1. I have no idea where to even start. I'm guessing I'd have to use substitution of some sort to find this antiderivative but I don't know how the square root would come into play. I know I could use x^3 as u and du would then be 3x^2 so it'd be 1/3 the integral of the square root of 1 which is just as complicated.

You could use trig sub.

$\displaystyle \int x^{3}\sqrt{x^{2}+1}dx$

Let $\displaystyle x=tan(t), \;\ dx=sec^{2}(t)dt$

Upon making the subs, we get:

$\displaystyle \int tan^{3}(t)sec^{3}(t)dt$

$\displaystyle \int tan^{2}(t)sec^{2}(t)(sec(t)tan(t)dt$

=$\displaystyle \int (sec^{2}(t)-1)sec^{2}(t)(sec(t)tan(t))dt$

Let $\displaystyle u=sec(t), \;\ du=sec(t)tan(t)dt$

$\displaystyle \int (u^{2}-1)u^{2}du$

$\displaystyle \frac{1}{5}u^{5}-\frac{1}{3}u^{3}$

Resub:

$\displaystyle \frac{1}{5}sec^{5}(t)-\frac{1}{3}sec^{3}(t)$

Resub from the beginning, $\displaystyle t=tan^{-1}(x)$

This gives:

$\displaystyle \frac{1}{5}(x^{2}+1)^{\frac{5}{2}}-\frac{1}{3}(x^{2}+1)^{\frac{3}{2}}+C$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~

An easier way may be to let $\displaystyle u=x^{2}+1, \;\ du=2xdx, \;\ \frac{1}{2}du=dx$

Then we get:

$\displaystyle \frac{1}{2}\int \sqrt{u}(u-1)du=\frac{1}{2}\left[\int u^{\frac{3}{2}}-\int u^{\frac{1}{2}}\right]du$

Can you go from there?. This way is easier if the trig sub is scary.

You can also use integration by parts, with $\displaystyle u=x^2$ and $\displaystyle dv=2x\sqrt{x^2+1}\,dx$:

$\displaystyle \int x^3\sqrt{x^2+1}\,dx$

$\displaystyle =\frac12\int x^2\left(2x\sqrt{x^2+1}\right)dx$

$\displaystyle =\frac12\int x^2\left[\frac23(x^2+1)^{3/2}\right]'dx$

$\displaystyle =\frac12\left[\frac23x^2(x^2+1)^{3/2}-\frac23\int2x(x^2+1)^{3/2}\,dx\right]$

$\displaystyle =\frac13x^2(x^2+1)^{3/2}-\frac13\int2x(x^2+1)^{3/2}\,dx$

$\displaystyle =\frac13x^2(x^2+1)^{3/2}-\frac13\left[\frac25(x^2+1)^{5/2}\right]+C$

$\displaystyle =\frac13x^2(x^2+1)^{3/2}-\frac2{15}(x^2+1)^{5/2}+C$

Just put $\displaystyle u^2=x^2+1$!

Originally Posted by galactus

You could use trig sub.

$\displaystyle \int x^{3}\sqrt{x^{2}+1}dx$

Let $\displaystyle x=tan(t), \;\ dx=sec^{2}(t)dt$

Upon making the subs, we get:

$\displaystyle \int tan^{3}(t)sec^{3}(t)dt$

$\displaystyle \int tan^{2}(t)sec^{2}(t)(sec(t)tan(t)dt$

=$\displaystyle \int (sec^{2}(t)-1)sec^{2}(t)(sec(t)tan(t))dt$

Let $\displaystyle u=sec(t), \;\ du=sec(t)tan(t)dt$

$\displaystyle \int (u^{2}-1)u^{2}du$

$\displaystyle \frac{1}{5}u^{5}-\frac{1}{3}u^{3}$

Resub:

$\displaystyle \frac{1}{5}sec^{5}(t)-\frac{1}{3}sec^{3}(t)$

Resub from the beginning, $\displaystyle t=tan^{-1}(x)$

This gives:

$\displaystyle \frac{1}{5}(x^{2}+1)^{\frac{5}{2}}-\frac{1}{3}(x^{2}+1)^{\frac{3}{2}}+C$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~

An easier way may be to let $\displaystyle u=x^{2}+1, \;\ du=2xdx, \;\ \frac{1}{2}du=dx$

Then we get:

$\displaystyle \frac{1}{2}\int \sqrt{u}(u-1)du=\frac{1}{2}\left[\int u^{\frac{3}{2}}-\int u^{\frac{1}{2}}\right]du$

Can you go from there?. This way is easier if the trig sub is scary.

I'm sorry, I don't understand where any of those numbers came from. Because there is no x in the function, they are all x's raised to powers, and none of your u's are raised to powers?


I haven't learned any of the methods any one else used yet, other than substitution to solve this problem. So if you guys, when helping me, could focus on the substitution method that would be appreciated.

Originally Posted by Krizalid

Just put $\displaystyle u^2=x^2+1$!

but then what would du be? And then what?

Originally Posted by Reckoner

You can also use integration by parts, with $\displaystyle u=x^2$ and $\displaystyle dv=2x\sqrt{x^2+1}\,dx$:

$\displaystyle \int x^3\sqrt{x^2+1}\,dx$

$\displaystyle =\frac12\int x^2\left(2x\sqrt{x^2+1}\right)dx$

$\displaystyle =\frac12\int x^2\left[\frac23(x^2+1)^{3/2}\right]'dx$

$\displaystyle =\frac12\left[\frac23x^2(x^2+1)^{3/2}-\frac23\int2x(x^2+1)^{3/2}\,dx\right]$

$\displaystyle =\frac13x^2(x^2+1)^{3/2}-\frac13\int2x(x^2+1)^{3/2}\,dx$

$\displaystyle =\frac13x^2(x^2+1)^{3/2}-\frac13\left[\frac25(x^2+1)^{5/2}\right]+C$

$\displaystyle =\frac13x^2(x^2+1)^{3/2}-\frac2{15}(x^2+1)^{5/2}+C$

I'm sorry, I don't want to solve it that way as I haven't learned that method yet

Originally Posted by fattydq

I'm sorry, I don't want to solve it that way as I haven't learned that method yet

$\displaystyle \int x^3 \sqrt{x^2 + 1} \, dx$

$\displaystyle u = x^2 + 1
$

$\displaystyle x^2 = u - 1$

$\displaystyle du = 2x \, dx$

set up the original integral for substitution ...

$\displaystyle \frac{1}{2} \int x^2 \sqrt{x^2 + 1} \cdot 2x \, dx$

substitute ...

$\displaystyle \frac{1}{2} \int (u - 1)\sqrt{u} \, du$

can you finish?

I solved it. Thanks guys!