1. ## Trigonometric Integration (tangent^4)

Hello, the help is very appreciated.

I'm asked to but not sure how to integrate the following:

tan(x)^4*dx

Many of our trig integration problems use the identities to simplify the integral, but I don't see how the identities could apply here.

Wolfram Alpha suggested using a (foreign to our class, I believe) reduction formula in the link below.

http://www.wolframalpha.com/input/?i=integrate+tan^4

I don't think we used this formula in class though. Are there any other ways of solving this?

2. Originally Posted by NBrunk
Hello, the help is very appreciated.

I'm asked to but not sure how to integrate the following:

tan(x)^4*dx

Many of our trig integration problems use the identities to simplify the integral, but I don't see how the identities could apply here.

Wolfram Alpha suggested using a (foreign to our class, I believe) reduction formula in the link below.

http://www.wolframalpha.com/input/?i=integrate+tan^4

I don't think we used this formula in class though. Are there any other ways of solving this?
Start by noting that $\displaystyle \tan^4 (x) = \tan^2 (x) \cdot \tan^2 (x) = \tan^2 (x)(\sec^2 (x) - 1)$.

3. I tried using the identity 1 + tan^2 = sec^2 and have gotten to

-x + integral sec(x)^4 = -x + integral tan(x)^2 + integral tan(x)^2*sec(x)^2

I tried then to continue with the same identity to get rid of the second term (with tan(x)^2), but it started flipping between integral of tan(x)^2 and sec(x)^2 indefinitely.

4. Originally Posted by NBrunk
I tried using the identity 1 + tan^2 = sec^2 and have gotten to

-x + integral sec(x)^4 = -x + integral tan(x)^2 + integral tan(x)^2*sec(x)^2

I tried then to continue with the same identity to get rid of the second term (with tan(x)^2), but it started flipping between integral of tan(x)^2 and sec(x)^2 indefinitely.
For the first integral substitute for tan^2(x) again. In the second, substitute u = tan(x).

Note: You're expected to know that $\displaystyle \int \sec^2 (x) \, dx = \tan (x) + C$.