 If our integrand has the form secant to a power tangent to a power, then if m is even, we can split off a secant squared and rewrite everything in terms of tangent. Well, if m and n are both odd, we can split off a secant x tangent x and rewrite everything in terms of secant. And that's because an even power of a trigonometric function can be converted into an even power of the co-function. But what if m is odd and n is even? What do we do then? So, for example, if we want to integrate secant cubed x dx, our usual trigonometric substitutions don't lead us anywhere useful, and so we'll try integration by parts. Why do we try integration by parts? Remember, when you're trying out something new, try the easy things first. So, we have three factors of secant, and since we know how to anti-differentiate secant squared x, we'll let that be rdv. So, dv is secant squared x dx, which means u has to be secant x. So, integrating and differentiating gives us, and so applying our integration by parts gives us. Now, let's simplify this integral a little bit. And because we have an even power of a trigonometric function, we can rewrite it in terms of the co-function. And we can expand and split this integral apart. Now, we end up with this integral of secant cubed again, but we can actually solve for integral of secant cubed and find. And so here is a formula for the integral of secant cubed. Now, we're not done yet, but here's the thing to notice. Notice that a formula for the integral of secant cubed introduced the integral of secant x. And this is an example of what's called a reduction formula, an integral involving a power that is expressed in terms of a similar integral involving a lower power. For example, suppose we have to find an expression for the integral of secant to the fifth expressed in terms of a lower power of secant. So we'll use integration by parts again. And again, since we know what the antiderivative of secant squared x is, we'll go ahead and make that be rdv, which means u has to be the remaining factors of secant. So integrating and differentiating, applying integration by parts. So again, we have the square of tangent. And so we can rewrite this in terms of the square of secant. So replacing, expanding, splitting the integral, and doing a little bit of algebra, we can then solve for secant to the fifth and get our reduction formula. And it looks like we'll always be able to do the step. So let's see if we can generalize it into a reduction formula. So let's see if we can find a reduction formula for the integral of secant to the nth dx. Since for secant cubed and secant to the fifth, we started with integration by parts. This time we'll use integration by parts. So again, we'll split off at factor of secant squared rdv. The remaining factors of secant will be our u, integrating and differentiating, applying the integration by parts formula. Again, we can replace the square of tangent with secant squared minus one, expanding, splitting the integral, and then doing a little bit of algebra to solve for secant to the nth. And so we have this formula that gives us the way to find the integral of secant to the nth. And remember, don't memorize formulas. You should be able to derive this as you need it, because all we really did here was to use integration by parts and the trigonometric identities. So for example, if I wanted to find the integral of secant to the fifth, so while there is a reduction formula, don't memorize it. And since we didn't waste our time memorizing the reduction formula, we'll use integration by parts to obtain. But wait, there's more. There's this integral of secant cubed x. Since we didn't waste our time with the reduction formula, we'll apply integration by parts to obtain. And so we've rewritten the integral of secant to the fifth in terms of the lowest possible power of an integral of secant. The reduction formula allows us to rewrite an integral involving an odd power of secant as another integral involving a lower odd power of secant. It would also work for an even power of secant, but we don't need it if we have an even power. But eventually, we need to find the integral of secant x dx. And what is that? We'll take a look at that next.