 In this video, we provide the solution to question number five for the practice final exam for math 1210. We're asked to evaluate the definite integral from negative 2 to 2 of 3x plus 1 quantity squared with respect to x. We're going to do this using the fundamental theorem calculus part 2. And in order to do that, we could do some type of u substitution like we could take u equals 3x plus 1 and go from there. That wouldn't be so bad. I'm just going to foil this thing out though. In which case, if you foil 3x plus 1 squared, you end up with 9x squared plus 6x plus 1 dx right here. For which then, if we calculate the antiderivative using the antiderivative version of the power rule, we end up with 3x cubed plus 3x squared plus x. And we're going to evaluate this from negative 2 to 2. So we're going to plug these numbers into the function here. So we're going to get plug it into the first one, right, x cubed. So we get a 2 cubed. That's going to be an 8 times that by 3. We get 24. Then we plug it into the next one. We're going to get x squared. It's going to become a 2 squared, which is 4 times that by 3. You're going to get a 12. And then you plug the next one in there. You're going to get a 2. We have to subtract from that if we plug a negative 2, right? So if you plug negative 2 in for x cubed, you're going to get a negative 8 times 3. It's a negative 24. If you plug it in for the x squared, you're going to get a positive 4 times that by 3, give you a positive 12. And then lastly, if you plug it in for the x, you're going to get a negative 2 there. Notice a little bit of simplification happens. You have 12 minus 12. Those cancel out. On the other hand, these other negatives will distribute and become positive, in fact. So adding things together, we get 24 plus 24, which is 48, 2 plus 2, which is 4. Adding those together, we get 52. And so we see that the correct answer would then be d52. Just as a side note, because of some of the way things canceled, when we had this thing expanded, because we have this symmetric interval from negative 2 to 2, it turns out that the odd part would just disappear and the even part would double up. So you could actually have ignored the 6x part if you wanted to, as we saw that with the 12s canceling out later on.