Question

\[\int {\sin^5 x dx} \]

Answer 1

So, should I start with the following?\[\int \left(\sin(x)\right)^4 \sin x dx\]

Answer 2

I think I might be able to use substitution rule in this one.

Answer 3

yes so u must create some cos from somewhere

Answer 4

Oh... wait... yes!\[\int (\sin^2(x))^2\sin xdx \]Now to use the identity\[\sin^2 x = 1 - \cos^2x \]

Answer 5

Right?

Answer 6

exactly

Answer 7

\[\int (1 - \cos ^2x)^2 - \sin x dx \]

Answer 8

Hmm... \(u = \cos x \) would work?

Answer 9

Yes ... let u = cos x

Answer 10

yes

Answer 11

Let's check.\[du = \cos x dx\]

Answer 12

Oops sorry for interrupting

Answer 13

Sorry, sinxdx ^^

Answer 14

Yeah!! Works!!

Answer 15

\[\int (1 - u^2)^2 du \]Right?

Answer 16

Oh, then to simplify (1 - u^2)^2?

Answer 17

sorry to butt in; but you can also use the reduction formulae; much faster.

Answer 18

Haven't heard of it, Mishi.

Answer 19

\[\int 1 - 2u^2 + u^4 du \]

Answer 20

lol, now I got it :)

Answer 21

Whom should I thank?

Answer 22

Thank mukushla ...

Answer 23

\[u - {2 \over 3}u^3 + {1 \over 5}u^5+c\]

Answer 24

OK, now to substitute back for u... I should be able to do that... thank you :D