Can the Reciprocal Rule Be Applied to Integrate Radical Functions?

[Stratosphere]

Homework Statement

$$\ y=\int x^{1/3} dx$$

Homework Equations

$$\ y=\int a x^{p/q} dx= a\frac{x^{(p/q)+1}}{(p/q)+1}+c$$

The Attempt at a Solution

I found the answer to it but I was wondering if you can also take the reciprocal of the exponent after adding the one to it and then multiply that by a, the theorem doesn't seem to imply that though. For all of the very low complex ones I'm doing it seems to work but I'm not sure about more advanced ones.

 

[HallsofIvy]

Presumably you know that
$$\int x^r dx= \frac{1}{r+1}x^{r+1}+ c$$
for r any real number.

That's exactly what your "relevant equation" says!

 

[g_edgar]

$$\int x^{1/3}\,dx = \frac{x^{\frac{1}{3}+1}}{\frac{1}{3}+1} +C = \frac{x^{4/3}}{4/3} +C = \frac{3}{4}\,x^{4/3} +C$$