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Find $\int (4 +\tan^22x) dx$

How do I go about integrating this, the $\tan^22x $ is where I'm stuck. How can I integrate this question?

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    $\begingroup$ Give the kid a break and stop the downvoting! $\endgroup$
    – user100463
    Apr 6, 2017 at 19:42

1 Answer 1

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$$\int\tan^2(2x)dx=\frac{1}{2}\int\tan^2(u)du=\frac{1}{2}\int\sec^2(u)-1du=\frac{1}{2}(\tan(u)-u)\\=\dfrac{\tan\left(2x\right)}{2}-x$$ Finally : $$\int (4 +\tan^2(2x)) dx=\dfrac{\tan\left(2x\right)}{2}+3x+C$$


Generally when you face some trigonometrics expressions with some $ax$ terms in them it is a good idea to make the change of variables $u=ax$. Then you can try to use some identities to make appear some know integrals : here we used $\tan^2(u)=\sec^2(u)-1$ because we know how to integrate $\sec^2$.
If you face a rational function of trigonometric functions you can take a look at this topic also for common useful substitutions : Evaluating $\int P(\sin x, \cos x) \text{d}x$

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    $\begingroup$ In general, when given a trig function with even power, attempt to use pythagorean theorem and known simple integrals. $\endgroup$ Apr 6, 2017 at 19:46
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    $\begingroup$ Alternatively one could write $\tan^2(2x) + 4 = \tan^2(2x) + 1 + 3 = \sec^2(2x) + 3$ and easily integrate that. Was going to post as an answer but I don't feel like it's different enough from this one. $\endgroup$
    – user307169
    Apr 6, 2017 at 20:02

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