Integration by parts and recursion formula

I'm supposed to convert the even powers of tangent to secant using a recursion formula, repeatedly if necessary, and then evaluate.

Here are to two problems:

∫ tan^2(x)sec(x) dx

∫ tan^4(x)sec(x) dx

Here is the recursion formula:

∫ sec^n(x) dx = (tan(x)sec^(n-2)(x))/(n-1) - ((n-2)/(n-1))⋅∫ sec^(n-2)(x) dx

I'm not sure how to go about this, so can someone please help me? Thanks!

Re: Integration by parts and recursion formula

Use the trigonometric relation now use the binomial expansion. Apply this to the above integrals.

Second integral becomes

Now use the recursion relationship on each integral.

Apply the same principle to the first integral as well.

Re: Integration by parts and recursion formula