Question

integration question.

Answer 1

I attached a pdf file for a solutions manual. On page 18 question number 21. I don't see why it goes from
\[-2t(5 - t)^{1/2} + 2 \int\limits_{?}^{?} (5 - t)^{1/2}dt \] TO
\[-2t(5-t)^{1/2} - 4/3(5-t)^{3/2} + c\]

I understand the anti derivative of \[2(5-t)^{1/2} = 4/3(5-t)^{3/2}\] But what i don't understand is why the 4/3 turned negative when the 2 was positive. Anyone know the answer?

Answer 2

I think youre not taking into account the -t when integrating

Answer 3

does it have to do with chain rule or something?

Answer 4

chain rule is only for differentiation

Answer 5

you want to use the substitution rule

Answer 6

u= the (5-t) im assuming?

Answer 7

chain rule. inside piece is \(5-t\)

Answer 8

yupper

Answer 9

i didn't even think about using substitution along with integration by parts

Answer 10

do i know to do that when if i were taking the derivative there would be a chain rule?

Answer 11

yeah if you were taking derivative it would be chain. substitution rule is basically in analogous technique in integration

Answer 12

thanks a bunch. Hope im ready for my final now

Answer 13

good luck!