 In this video, we provide the solution to question number six for practice exam number one for math 1220, in which case we have to compute the indefinite integral cosecant theta times cotangent theta plus two times e to the two theta with respect to theta. Now I'm going to treat this as two separate problems. We're going to look for an antiderivative cosecant cotangent, and then we're going to deal with two e to the two theta there. So with the first one, this one actually resembles a derivative that we know. So just as a reminder, we might know from like calculus one that the derivative of theta, excuse me, of secant theta is equal to secant tangent. Now this is when we remember pretty well. The reason I mentioned this is because the derivative of cosecant is almost the exact same thing. The derivative is going to be a cosecant theta, a cotangent theta. The only difference is because it's a cosecant as opposed to secant, there's a negative sign out in front. So when we look at this one right here, we have the cosecant, we have the cotangent, we don't have the negative. And so the antiderivative is like, oh, okay, this is not the derivative of cosecant, it's the derivative of negative cosecant. So the derivative of the first one will be negative cosecant, the antiderivative will be negative cosecant. Now as we look at the next one, two e to the two theta, if you took the derivative of e to the two theta with respect to theta by the chain rule, this gives us two e to the two theta. That's exactly what we have right here. So we see that the second piece is going to be e to the two theta. Don't forget your constant, you do need that plus c for full credit, and then we get that the answer would be negative cosecant theta plus e to the two theta plus c.