Is there a general method, some sort of algorithm, for evaluating integrals? I find myself having trouble knowing what method I am supposed to use at what point.
For example, how does one go about something like $$\int\frac{x^3}{x^2+1}dx$$ or $$\int\frac{e^x +1}{e^x - 1}dx$$
What am I looking for when I am not sure how to tackle a complex integral?
On
No, there's no general method for evaluating an arbitrary integral. Unlike differentiation, the reverse operation isn't algorithmic. So calling it the integral calculus is, properly speaking, a misnomer, for it's not a calculus like differentiation, or elementary arithmetic, say.
So, the thing about evaluating integrals is practice, experience and ingenuity. You make clever substitutions, educated guesses, etc. This, again, only emerges by much practice.
For the first you need to write the following.$$\frac{x^3}{x^2+1}=\frac{x^3+x-x}{x^2+1}=x-\frac{x}{x^2+1}$$ because you know that $$\left(\ln(x^2+1)\right)'=\frac{2x}{x^2+1}.$$ For the second you need to write the following. $$\frac{e^x+1}{e^x-1}=\frac{2e^x-e^x+1}{e^x-1}=\frac{2e^x}{e^x-1}-1$$ because you know that $$\left(\ln|e^x-1|\right)'=\frac{e^x}{e^x-1}.$$