Question

Evaluate sin(5x)dx from pi/3 to pi/2

Answer 1

\(\color{black}{\displaystyle \int_{\pi/3}^{\pi/2}\sin(5x)dx }\)

Answer 2

This ??

Answer 3

Yes

Answer 4

Are you relevantly new to integration?

Answer 5

Yes we're just learning about it Calc 1

Answer 6

Then, set \(\color{black}{\displaystyle u=5x }\)

Answer 7

If u=5x, then du/dx=5. Right?

Answer 8

So, multiplying both sides times dx, we get:
du = 5dx

Answer 9

then, solving for dx, we get dx=(1/5)du.

Answer 10

Yes

Answer 11

\(\color{black}{\displaystyle \int_{\pi/3}^{\pi/2}\sin(5x)dx }\)

\(\color{black}{\displaystyle u=5x }\)
\(\color{black}{\displaystyle (1/5)du=dx }\)

then, change your limits to fit variable \(u\). (Know how to do this?)

Answer 12

What Solomon is doing here is using the substitution u=5x. From that he gets du=5dx.
Make sure you're comfortable with this before progressing further.

Answer 13

Plug pi/2 and pi/3 into u= equation?

Answer 14

Yes!

Answer 15

But that step comes AFTER you've done the indicated integration. Have you done it?

Answer 16

So if \(\color{black}{\displaystyle x=\pi/2 }\) then \(\color{black}{\displaystyle u=5(\pi/2)=5\pi/2 }\)
and for \(\color{black}{\displaystyle x=\pi/3 }\) you get \(\color{black}{\displaystyle u=5(\pi/3)=5\pi/3 }\)

Answer 17

So having made a substitution
\(\color{black}{\displaystyle u=5x }\)
\(\color{black}{\displaystyle (1/5)du=dx }\)

and having accounted for new limits of integration, what is your NEW integral?

Answer 18

1/5 sin(u)du from 5pi/3 to 5pi/2?

Answer 19

YES, AWESOME!!!

Answer 20

\(\color{black}{\displaystyle \frac{1}{5} \int_{5\pi/3}^{5\pi/2} \sin(u)du }\)

Answer 21

Can you evaluate this integral?

Answer 22

What is the integral of sin(u) ?
(What function, when differentiated, has a derivative of sin(u) ??)

Answer 23

-cosu

Answer 24

Yes, fabulous!

Answer 25

\(\large \color{black}{\displaystyle \frac{1}{5} \left.\int_{5\pi/3}^{5\pi/2} \sin(u)du=-\cos(u)\right|_{u=5\pi/3}^{u=5\pi/2} }\)

Answer 26

If you want, you may do the following:
\(\large \color{black}{\displaystyle \frac{1}{5} \left.\left.\int_{5\pi/3}^{5\pi/2} \sin(u)du=-\cos(u)\right|_{u=5\pi/3}^{u=5\pi/2}=\cos(u)\right|^{u=5\pi/3}_{u=5\pi/2} }\)

Answer 27

in other words, multiply the whole negative 1, and reverse the bounds.
(It still remains equivalent if you think about it.)

Answer 28

the whole `thing by` -1.

Answer 29

Alright that may be easier

Answer 30

You can evaluate it either way, though, I am pretty sure;)

Answer 31

go ahead, and take your time:)

Answer 32

Kayders and Solomon: Keep that (1/5). Looks like the (1/5) was inadvertently dropped.

Answer 33

Is it 1/10???

Answer 34

True, mathmale.

Answer 35

Kayders, awesomeness!

Answer 36

Thank you so much @SolomonZelman :)

Answer 37

No problem, yw!