How can I get better at integration? I'm an A-Level student in the UK, and the new school year is about to start. I'm concerned that my integration skills aren't as good as they should be, and many people in my class also struggle with integration (but not differentiation). I've heard the saying 'differentiation is a science, integration is an art' and I must agree. My problem is that I can never tell which rule to use - substitution, integration by parts or using some form of the 'special' integrals such as
$$\int{}\frac{f'(x)}{f(x)}dx=\ln|f(x)|+c$$
or
$$\int{}\frac{1}{a^2+x^2}dx=\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)+c$$
How anybody came up with integrating the Gaussian Integral is beyond me. So how can I improve?

*

*Practice. Stewart's (RIP) Calculus is great.


You can find a lot of practice problems online:
http://www.sosmath.com/cyberexam/calculus/E004/E004.html
http://math.feld.cvut.cz/mt/txtd/3/txe3dc3.htm
You can even look up tricky integration problems:
http://www.intuitive-calculus.com/integral-by-tricky-substitution.html
http://echochamber.me/viewtopic.php?t=75085


*Note down ones with which you had difficulty and review those later.


P.S. You can always check your answers on WolframAlpha
