 Let's take a look at the solution for question one on our midterm exam for calculus 1220 And this one we want to evaluate the integral from negative 2 to 2 of 3x plus 1 quantity squared This one I would probably just suggest that we just foil out the 3x plus 1 squared there Upon doing so you're gonna get 3x times 3x, which is 9x squared Then you're gonna get a 3x times 1 and then another 3x times 1 so that gives you a 6x and then 1 times 1 which is 1 Compute an antiderivative When you take the antiderivative of the 9x squared you're gonna get 9x cubed over 3 plus 6x squared over 2 Plus x as you go from negative 2 to 2 The coefficients do simplify a little bit you get 3 goes into 9 leaving a 3 2 goes into 6 leaving also a 3 and So plug these things in there you plug in the 2 You're going to well, so we'll write the simplified version there You're gonna get 3x cube plus 3x squared plus x as you go from negative 2 to 2 So when you plug these things in there, you're gonna get 3 times 2 Cubed which that's gonna be an 8 You're adding to that 3 times 2 squared, which is 4 plus a 2 so track from that We're gonna put a negative 2 in there and so you're gonna get 3 times negative 8. You're going to get 3 times positive 4 and then lastly you're gonna get a negative 2 Like so and so notice some cancellation that happens here In the case you have a 3 times 4 which is a 12 that's gonna cancel with the 3 times 4 which is 12 as well So that cancels out but on the other ones when you distribute this negative sign actually makes them positive So things are gonna double up so 3 times 4 Sorry 3 times 8 is 24, so you're gonna get 2 times 24 for the first one there and then 2 2 plus 2 So 2 times 2 which is a 4 so you end up with 48 and 4 which adds up to be 52 Thus leading us to choose option D and So it's just meant to be a fairly straightforward calculation in that regard But as a sort of a comparison, I did also want to mention that if you came back to this step right here Since we were integrating across 2 and negative 2 you could use symmetry to help you out here You could break this thing up into the even part Which would be 9x squared plus 1 dx And then you could do the odd part negative 2 to 2 of the 6x dx When you integrate an odd function across in Symmetric interval just cancels out and then so that so the integrating the 6x just cancel that like we saw up here Right, but then also when you integrate across the even part You could have just turned 2 times the integral from 0 to 2 of 9x squared plus 1 dx And so then if you had done finish that calculation everything else would be basically the same Using symmetry here makes it a little bit cleaner I showed you all the details nonetheless and that's going to give us the solution to question number one