1. help with an integral

I'm having a lot of trouble with this integration. I'm supposed to simplify the integral so that it looks like one of the equations on a table of integrals, which then can be used to find the integral.

Integral of (x^3)*sqrt[(4x^2) -(x^4)]

Thanks.

2. Put $\displaystyle u=x^2,$ but that won't solve the problem, you need to manage of the other integrals.

3. $\displaystyle \int x^{3}\sqrt{4x^{2}-x^{4}}dx$

Rewrite as:

$\displaystyle \int x^{4}\cdot\sqrt{4-x^{2}}dx$

Now, you can use various methods, but trig sub is an option.

You can look at the tables or try to do it yourself.

Make the sub $\displaystyle x=2sin(t).\;\ dx=2cos(t)dt$

We get $\displaystyle 64\int sin^{4}(t)dt-64\int sin^{6}(t)dt$

$\displaystyle 64\int sin^{4}(t)dt=8\int cos(4t)dt-32\int cos(2t)dt+24\int dt$

Also $\displaystyle \int sin^{6}(t)dt=\int\frac{5}{16}dt-\frac{15}{32}\int cos(2t)dt+\frac{3}{16}\int cos(4t)dt-\frac{1}{32}\int cos(6t)dt$

See?. You can break it all up into easier integrals and go from there.

Of course, as I said, this is one way.