A list of techniques to try when confronted with an integral?

Question

I didn't solve any integrals for 2 months, and now I have lost most of my reflexes.

What is a good checklist of techniques to try, and in which particular order ? The purpose is to explore every route each time, and forget none of them.

Answer 1

Maybe you can try these steps in a similar order:

1. Trying the simple rule for monoms $\int a\cdot x^n \text{d}x=\frac{a}{n+1}x^{n+1}$ for $n\neq -1$. (For $n=-1$: $\int\frac{1}{x}\text{d}x=\ln x$)
2. Knowing antiderivatives of many functions (special functions). For example $\int \cos x\text{d}x=\sin x$
3. Trying to substitute (For example $\int \cos(2x)\text{d}x=\frac{1}{2}\int \cos x'\text{d}x'$). Often a completion of a square before substituition is very useful ($ax^2+bx+c=x'^2+c'$ then substitute $x\to x'$)
4. Integration by part $\int f(x)g(x) \text{d}x = F(x)g(x) - \int F(x)g'(x) \text{d}x$. with $F(x)$ is the antiderivative of $f(x)$