Question

Evaluate the integral

Answer 1

\[\int\limits_{}^{} \sin^2(\pi x) \cos^5(\pi x)\]

Answer 2

so I make this \[\int\limits_{}^{}\sin^2(\pi x) (1-\sin^2(\pi x))^2 \cos(\pi x)\]

Answer 3

would I then use U substitution?

Answer 4

or can I factor out the PI?

Answer 5

@iambatman

Answer 6

hmmm you second comment what yo did is incorrect

Answer 7

how about you do this:
\[\int \sin^2\pi x \cos ^5\pi x=\int \sin^2 \pi x (1-\sin^2 \pi x)^2\cos \pi xdx\]
use U sub
\[u=\sin \pi x \Longrightarrow du=\pi \cos \pi x dx\]

Answer 8

actually you did good just looked it again :)

Answer 9

okay I was close though lol

Answer 10

okay let me give this a try

Answer 11

yeah pretty close :)

Answer 12

why is du = pi cos pix

Answer 13

chain rule

Answer 14

do you mind writing out how you got there I am terrible with trig

Answer 15

\[(f(g(x)))'=g'(x)f'(g(x)))\]
this is the chain rule

Answer 16

suppose \[u=\pi x\]
then we have \[\sin u\]
when we differentiate that with respect to x
we should have \[\frac{d}{dx}\sin u=\frac{d}{dx}(u)\frac{d}{du}\sin u=u'\cos u\]
what is u' its just pi
\[u=\pi x \Longrightarrow u'=\pi \]
so we get
\[\pi \cos \pi x\]

Answer 17

okay thanks, so I get this now\[\frac{ 1 }{ \pi }\int\limits_{}^{}u^2(1-u^2)du\]

Answer 18

no! you forgot one thing
think about it more

Answer 19

close :)

Answer 20

oh the ^2

Answer 21

yeah good catch you forgot the square

Answer 22

can you do it now

Answer 23

Let me give it a shot

Answer 24

go :)

Answer 25

Can you check my distribution \[(u^2 - 2u^4 + u^8) \]

Answer 26

u^6 not u^8

Answer 27

the rest is good

Answer 28

okay so now I integrate that

Answer 29

yep

Answer 30

\[\frac{ 1 }{ \pi }(\frac{ 1 }{ 3 }u^3-\frac{ 2 }{ 5 }u^5+\frac{ 1 }{ 9 }u^9)\]

Answer 31

+ c

Answer 32

forgot that is a six

Answer 33

i said the last term was u^6 not u^8 lol

Answer 34

yah lol

Answer 35

besides that

Answer 36

good job :)

Answer 37

just go back to sinpix don't leave it u

Answer 38

\[\frac{ 1 }{ \pi }(\frac{ 1 }{ 3 }(\sin \pi x)^3-\frac{ 2 }{ 5 }(\sin \pi x)^5 + \frac{ 1 }{ 7 } (\sin \pi x)^7 + c\]

Answer 39

now distribute the pi

Answer 40

good job

Answer 41

whatever you like hehe
i prefer it that way

Answer 42

Awesome thanks for your patience!

Answer 43

you are awesome actually hehe
and no problem

Answer 44

haha thanks

Answer 45

-_^ keep it up, i think you are good at this
work more of this for mastery

Answer 46

will do!