Question

How to evaluate the integral of:

Answer 1

\[\int\limits_{0}^{1} \sin(3\pi(t))dt\]

Answer 2

I know hot to use FTC and U substitution, but I'm having trouble with knowing when to use which..

Answer 3

anti derivative is easy enough

Answer 4

*how

Answer 5

\[-\frac{1}{3\pi}\cos(3\pi t)\]

Answer 6

a mental u-sub

Answer 7

Where did the (1/3pi) come from? I get why it's negative cos(...)

Answer 8

what is the derivative of
\[-\cos(3\pi t)\]

Answer 9

sin(3(pi)t)?

Answer 10

oh no !

Answer 11

you need the chain rule for that one

Answer 12

OH does this involve chain rule?

Answer 13

Okay!

Answer 14

let me know when you get
\[3\pi \sin(3\pi t)\]

Answer 15

yeah, I got that!

Answer 16

but that isn't what you want is it? you just want
\[\sin(3\pi t)\]

Answer 17

Oh okay, I understand now. So in this case, would u just be the entire function? sin(3pi(t))?

Answer 18

u shmoo
you don't want
\[3\pi \sin(3\pi t)\] you just want \[\sin(3\pi t)\] so divide ! make it
\[-\frac{1}{3\pi}\cos(3\pi t)\] and it will work out nicely
it is a u - sub you do in your head

Answer 19

Okay. So generally, how can I look at an integral and know which method to use?