Powers or Trig Functions Question - Integral

[Shkolnikoff]

Homework Statement

Integral of : (sec x)^5 (tan x)^3 dx

Homework Equations

I am having trouble substituting correctly for this equation and i can't get it to work. I believe it has to do with trig powers ?

If anyone could help be step by step. or even just a hint on how to solve it that would be great.

Thanks in advance !

 

[Gib Z]

Welcome to PF!

My Hint would be to rewrite the tan cubed term in terms of a square and a linear, then change the square term with a Pythagorean identity, expand and see where that takes you.

 

[konthelion]

You should probably memorize this rule for integrating trigonometric integrals

For any $$\int{tan^{m}xsec^{n}xdx}$$

If n is even, substitute u = tan(x)
If m is odd, substitute u = sec(x)
If m is even, n is odd, reduce to powers* of sec(x)
Also, using the identity $$1+tan^{2}x=sec^{2}x$$ is helpful in all three cases

 

[Shkolnikoff]

Thank you very much for all your help. I just finished solving the problem.

Its a lot easier using the hints you provided! Thanks!