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