What is the antiderivative of #f(x)=tan (6x)#?

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Answer 1 · 2018-03-25T16:39:35

#1/6ln|sec(6x)|#

or if you prefer:

#-1/6ln|cos(6x)|#

The integral of #tan(x)# is known, and it can be found on most formula sheets with integrals of basic trig functions:

#inttan(x)dx=ln|sec(x)|#

If you want a challenge, try to prove that this is true by using the identity:

#tan(x)=sin(x)/cos(x)#

Then integrate using #u#-subtitution with #u=cos(x)#.