How can I get better at integration?

Question

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?

Answer 1

Practice. A lot.

Do as many examples and exercises as you can. There are lots of websites offering integration practice etc.

A rule my lecturer always told me is that with every integral you get a free exercise in differentiation to check your answer.

Apart from that, just practice a lot and it will come to you.

Answer 2

1. 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

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

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