Question

how do you compute this indefinite integral
∫tan⁡(2x)dx
thanks in advance

Answer 1

Rewrite this as \[\int\limits \sin(2x)/\cos(2x)\]
Now use the substitution \[y = \cos(2x)\]\[dy=-2\sin(2x)\]
To arrive at an integral of the type \[-(1/2(\int\limits dy/y\]
As you probably know, that equals \[-(1/2)\ln|y|+c =-(1/2)\ln|cos(2x)|+c\]

This site has a great resource on that http://math.feld.cvut.cz/mt/indexed.htm

Answer 2

thanks alot for the steps :) i had a question,how do you know which term to substitute ? can you substitute sinx instead of cosx?

Answer 3

It wouldn't work - you've got cosine in the denominator and you wouldn't eliminate it like that. With integrals (like with many things), practice is the answer. With trigonometric functions, rewriting in terms of sine and cosine and doodling with identities usually helps. There are also automatic substituions you can use (tan substitution), often they create even greater mess, but they can also be very helpful. The site I sent you is absolutely fantastic for that!