(Yes, I’m bored. So why not completing my series.)

Antiderivative of tangent

First, remember that , so by chain rule we obtain . By the (first) fundamental theorem of calculus, derivative and antiderivative are two inverse operations, thus we obtain .

Also, recall that and . Don’t tell me you don’t know this; if you don’t, then stop reading.

To the real meat.

(!)

Letting , we have (!). Thus,

Which by our lemma above is equal to

(Hint: Wikipedia is your friend.)

Usually I stop here, but you can also simplify it to involve only one trigonometric function:

(!)

Call . So

(remember ?)

Thus .

Now, what happens to …

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*Related*

You can try Half tangent angle substitution as well which is more common and standard.

For you bonus secant cube question: apply integration by parts.