Why Is Substitution Failing in Integrating This Function?

[songoku]

Homework Statement

Find
$$\int \frac{x^3}{\sqrt{16x-x^8}}dx$$

Relevant Equations

u - substitution

trigonometry substitution

I tried using substitution $u=\sqrt{16x-x^8}$, didn't work

Tried factorize $x$ from the denominator and then used $u=\sqrt{16-x^7}$, didn't work

Tried using $u=x^4$ also didn't work

How to approach this question? Thanks

 

[romsofia]

Tried factorize $x$ from the denominator and then used $u=\sqrt{16-x^7}$, didn't work

This was the right approach, but your u-sub is too fancy. Use a different u-sub from the expression $\sqrt{x(16-x^7)}$.
It falls into place after that. In this problem you'll have to bring something back from outside the square root (which you'll see once you use the right U-sub), a complete the square, and a trig sub.

 

[songoku]

Thank you very much for the help romsofia

 

[docnet]

$u=\frac{x^{7/2}}{4}$ looks promising