Can you do integral of sec^5(x)tan^2(x) without reduction formula?

[aselin0331]

Homework Statement

Can you do integral of sec^5(x)tan^2(x) without reduction formula?

Homework Equations

The Attempt at a Solution

If so, would it be integration by parts? I tried splitting it up in way too many ways to post them on here.

Thanks for any hints!

 

[Mark44]

Homework Statement

Can you do integral of sec^5(x)tan^2(x) without reduction formula?

Homework Equations

The Attempt at a Solution

If so, would it be integration by parts? I tried splitting it up in way too many ways to post them on here.

Thanks for any hints!

First, get everything in terms of sec(x).
$$\int sec^5(x)tan^2(x)dx = \int sec^7(x) - sec^5(x)dx = \int sec^7(x)dx - \int sec^5(x)dx$$

Use integration by parts on both integrals on the right. For the first u = sec⁵(x) and dv = sec²(x) dx. For the second integral, u = sec³(x) and dv = sec²(x) dx.

I haven't taken this all the way through, but I'm reasonably sure it will work. You will probably need to solve for your integral algebraically.

 

[aselin0331]

You're fantastic! I was staring at Sec^7 and sec^5 for a while and that just clicked in!

Thanks

 

[Mark44]

Aw shucks!