# Antiderivative of product of trig functions

## Homework Statement

Find the antiderivative of $$sin^4(x)tan^2(x)$$

## Homework Equations

Trig identities I may have overlooked.

## The Attempt at a Solution

I tried writing the integrand in terms of sin and cos but that didn't seem to lead anywhere. I tried integration by parts since the antiderivative of $$tan^2(x)$$ is $$tanx - x + C$$, but that approach lead to a more difficult integral that might have been solved using a messier integration by parts.

This isn't really homework since I'm in a theoretical calc course and we begin the theory of integration next quarter (in fact, we're on break). Anyways I used to be pretty good with elementary terms integration but I'm stuck. Could someone suggest a good first step? Thanks.

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Mark44
Mentor
I think this will go with integration by parts.
Try u = tan^2(x)sin^3(x) and dv = sin(x)dx.

This gave me an unwieldy-looking integral, but after simplifying to sines and cosines, I was left with $$\int \frac{sin x}{cos^2 x} dx$$ and $$\int sin^4 x dx$$. There were constant multipliers that I have omitted.

The first integral is an easy one and the second isn't too hard.
Hope that helps.

Another potential way to do this integral is by converting everything to cosines because there is a cosine downstairs:

$$\int \sin^4x\,\tan^2x\,dx = \int\frac{(1-\cos^2 x)^3}{\cos^2 x}\,dx = \int \frac{1 - 3\cos^2 x + 3\cos^4 x - \cos^6 x}{\cos^2 x}\,dx$$​

Each of the resulting integrals should be fairly simple to solve.

Thank you both for the suggestions.

Mark44, did you arrive at those two integrals after applying integration by parts once? I got sin sin^4(x) term but the the one before that (I think it was sin^4(x)/cos^2(x) or something). I'll try it again though.

Tedjn, I thought of the very first equality you posed but did not think to expand it. That is quite a neat trick, i.e., decrease the powers by expansion. A few expansions and some computation and it worked out well. Thanks.

Mark44
Mentor
Thank you both for the suggestions.

Mark44, did you arrive at those two integrals after applying integration by parts once?
Yes.
I got sin sin^4(x) term but the the one before that (I think it was sin^4(x)/cos^2(x) or something). I'll try it again though.

Tedjn, I thought of the very first equality you posed but did not think to expand it. That is quite a neat trick, i.e., decrease the powers by expansion. A few expansions and some computation and it worked out well. Thanks.