8.3 Integration with Trig Powers Part 2- Secants and Tangents

Integrals Involving Secant and Tangent Powers Recall the identity: Ex. 1 Could I do a u-sub now?

Ex. 2 What could we “pull out” of here to get a u-sub?

Tips for working with Secant and Tangent Powers: 1) If secant is even and positive, split up and save a secant-squared term and use identities to convert the remaining factors to tangents. 2) If tangent is odd and positive, save a secant-tangent factor and convert the remaining factors to secants. 3) If there are no secants, and tangent is even and positive, convert a tangent-squared factor to a secant-squared factor and expand. 4) If there are no tangents, and secant is odd and positive, try integration by parts. 5) If all else fails, try converting to sines and cosines.

Ex. 3 Let’s use the identity:

Ex. 4 Trig Identity:

We need a secxtanx in here.. Ex. 5 We need a secxtanx in here.. Use an identity to convert to all secants:

Ex. 6 Let’s try using integration by parts. This we can integrate using new techniques!

Since none of our “tips” apply, let’s change to Ex. 7 Since none of our “tips” apply, let’s change to sines and cosines and see what happens: