(High school) How to become an expert on integration? It is very hard for me sometimes to find a right method to do an integration, for example, whether to use by parts method or substitution. I always find the questions I was given on an exam very hard to do without spending much time experimenting on it (I have to skip otherwise I will run out of time). An example question would be to integrate an arcsin(x).
So I wonder if there's any site that provides solutions for all different kinds of integration or something that will help me to recall instantly the method I need to use to integrate an equation? or any advice would be appreciate!!
 A: 1) You need to know the tables of derivatives/primitives of elementary functions by heart, flawlessly. (Recall that powers yield powers, polynomials yield polynomials, exponential and $\sin/\cos$ yield themselves.)
2) Recognize factors of the integrand that you can integrate, i.e. write $f$ as $u'.v$.
2a) In some cases, $v$ can be expressed as a function of $u$. For example, $x.e^{x^2}=(x^2)'.e^{x^2}$ hints the substitution $u=x^2$. (Unessential constant omitted.)
2b) In other cases, it may appear that $u.v'$ would be somewhat simpler to integrate: try integration by parts. For example, in $x.\cos x$, you can integrate both $x$ or $\cos x$. Deriving the other factor will give $\sin x$ and $1$ respectively, resulting in integrands $x^2.\sin x$ (worse than before) or $\sin x$ (immediate).
2c) If you can't find an integrable factor, try this trick: $\int f(x)dx=\int x'f(x)dx=xf(x)-\int xf'(x)dx$.
In the case of $\arcsin x$, you won't find a suitable primitive in the tables (but note that the derivative is $\frac1{\sqrt{1-x^2}}$). You don't see an integrable factor either. Then observe that by using rule 2c, you will get rid of the $\arcsin$ !
Now you have to solve $\int \frac x{\sqrt{1-x^2}}dx$. You can integrate $x$ as $x^2$, a candidate for the $u$ function. But an even better $u$ candidate is $1-x^2$, as appearing under the radical, also having $x$ for derivative. It will yield $\int \frac{dx}{\sqrt x}$, found in the tables.
For this case, another strategy can be attempted: get rid of the annoying $\arcsin$ function from the beginning by subsitituting $x$ with $\sin x$. You integrand will turn to the form $\arcsin(\sin x).\cos x=x.\cos x$.
