how do you know when you fully integrated something

Question

How do you know when you've fully integrated something?

For instance, the integral:

$$\int \tan^5(x)\sec^3(x)\,\mathrm{d}x$$

Firstly, how do I integrate this, and secondly, how do I know if the integral is fully done.

Answer 1

I integrate it observing that$$\int\tan^5(x)\sec^3(x)\,\mathrm dx=\int\frac{\sin^5(x)}{\cos^8(x)}\,\mathrm dx=\int\frac{\sin(x)\bigl(1-\cos^2(x)\bigr)^2}{\cos^8(x)}\,\mathrm dx$$and using the substitution $\cos(x)=y$ and $\sin(x)\,\mathrm dx=\mathrm dy$. And I check my answer computing its derivative and seeing wither or not I grt $\tan^5(x)\sec^3(x)$.

Answer 2

1) Try something like $u=\sec(x)$, recall $$du=\sec(x)\tan(x)dx$$ 2) The integral "is fully done" when the symbol $\int $ no longer appears.