Question

the question is in the picture below

Answer 1

do we have to integrate this

Answer 2

yes

Answer 3

You basically want to set \(F(x) = 0\) so that you can find the roots.

Answer 4

Roots let you know when the sign will change.

Answer 5

It's an easy integral though. \[
\int\limits_0^x\sin(2t) ~dt = \frac12 \int\limits_0^x\sin(2t) ~2dt= \frac12 \int\limits_0^x\sin(2t) ~d(2t) = \frac12\left(-\cos(2t)\Bigg|_0^x\right)
\]

Answer 6

thats it? oh wow. so its just 1/2cos(2x)-1/2cos(2*0) so it'd be -1/2cos(2x)-1

Answer 7

sorry i left some signs out

Answer 8

then from there i just plug in the pi? to find F(pi)

Answer 9

I would say: \[
F(x) = \frac 12-\frac 12\cos(2x)
\]

Answer 10

Yes.

Answer 11

isn't the cos (0) just 1? or where'd that extra 1/2 come from?

Answer 12

One of the \(\cos\) simplifies, the other doesn't due to \(x\).

Answer 13

ohh okay let me plug in pi, hang on

Answer 14

so its just 0

Answer 15

thanks! got it right! :)