 Integrals involving trigonometric functions often rely on trigonometric identities. The main identities you should know are the definitional identities that give you tangent, secant, and cosecant in terms of sine and cosine, and the Pythagorean identities. It will be very useful to know the co-function identities for cosine, cotangent, and cosecant in terms of sine, tangent, and secant, and conversely, as well as the angle sum and difference identities. But what if you forget them? The thing to keep in mind is that a problem exists whether or not you know how to solve it. There's a lot of trig identities, and you can get bogged down in memorizing a whole bunch of things, but remember, understand concepts don't memorize formulas. So for example, let's find the integral of cosine squared. Now there's a reduction formula for powers of sine and cosine, but if you don't remember it, what can you do? Yeah, that's not that useful. So the thing to keep in mind is there's really only two integration techniques. Substitution and integration by parts. All else is algebra, which will include some trigonometric identities. So there's no obvious substitution, so let's try integration by parts. We note that cosine squared is cosine times cosine, so we'll let u be cosine and dv be cosine, and we'll differentiate and integrate, and integration by parts will give us. Now you can't completely abandon the identities, but if you know the definitional identities and the Pythagorean identities, you'll be able to do most things. So our Pythagorean identity can be used to convert this sine squared in to, and we can simplify it and rewrite our integral as, and we can evaluate the first integral. And notice we have a cosine squared theta on both sides of the equation. So we can solve for the integral of cosine squared theta, or let's try the integral of sine to the fourth. Since we know the antiderivative of sine theta will make the choices for u and dv, differentiating and integrating, and so integration by parts will give us. Using the Pythagorean identity, we can rewrite cosine squared and simplify. Now we have the integral of sine to the fourth on both sides, so we can solve for it. Now we could find sine squared theta d theta using integration by parts again, but we just found the integral of cosine squared theta d theta, so let's use the Pythagorean identity to rewrite this as, and we can make use of our integral of cosine squared theta d theta, and using our previous result gives us. Since the traditional approach for integrals like this uses the reduction formulas, your answer may be different. So how can you check? First of all remember, in the real world you can't look up the answers, but you still need to check them. So according to a computer algebra system, the integral of sine to the fourth is this, but we found this, and let's verify the answers are correct. So the obvious thing to try is we could differentiate both answers and verify their derivatives are equal to sine to the fourth, but this would require using a lot of trigonometric simplifications. However, if we're already using a computer algebra system, we can also rely on a theorem, the antiderivatives of a function differ by a constant, and so we can find their difference. And since the difference is a constant, then both are antiderivatives. Or we can graph the two functions. And we see the graphs of the two functions appear to be the same, so they're probably both correct.