Integrate cos^3xdx - First Clue: Substitute (1-(Sinx)^2)cosxdx

[lastlaugh]

Any suggestions on how to integrate
\int cos^3xdx
My first clue would be to substitute
(1-(Sinx)^2)cosxdx but any clue to go after this?

 

[Hurkyl]

Well, do any of the techniques you've learned apply to

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

?

 

[himanshu121]

Put sinx=t => cosx dx=dt

so u have int(1-t²)dt=t-t³/3

 

[lastlaugh]

thanks for the help. This trig integration stuff is killer

OK the reason I asked was because I have a lab assignment where I have to calculate the powers of \int cos^n[x]dx where n is odd numbers from 3-13.
So can someone please look over my work for \int cos^5[x]dx and make sure it looks valid:
First I substituted (1-sin^2[x])^2 for cos^4[x] which gives me:
\int (1-sin^2[x])^2 cos[x]dx
Then I let u=sinx meaning du=cosxdx
Giving me:
\int (1-u^2)^2du
Which I simplfied to give me:
\int u^4 du - \int 2u^2 du + \int du
Which gives me
(u^5)/5-(2u^3)/3 + u

Anyone see any glaring errors?
Thank you in advance.

 

[PrudensOptimus]

Originally posted by lastlaugh
thanks for the help. This trig integration stuff is killer

OK the reason I asked was because I have a lab assignment where I have to calculate the powers of \int cos^n[x]dx where n is odd numbers from 3-13.
So can someone please look over my work for \int cos^5[x]dx and make sure it looks valid:
First I substituted (1-sin^2[x])^2 for cos^4[x] which gives me:
\int (1-sin^2[x])^2 cos[x]dx
Then I let u=sinx meaning du=cosxdx
Giving me:
\int (1-u^2)^2du
Which I simplfied to give me:
\int u^4 du - \int 2u^2 du + \int du
Which gives me
(u^5)/5-(2u^3)/3 + u

Anyone see any glaring errors?
Thank you in advance.

WRONGGGGGGGGGGGGGGGGGGGG

is (1-u^2) du.

 

[Hurkyl]

Looks right to me. (except you should substitute back what u is)

 

[himanshu121]

Originally posted by PrudensOptimus
WRONGGGGGGGGGGGGGGGGGGGG

is (1-u^2) du.

How it is wrong

Moreover General way will be to form a reduction formula

 

[lastlaugh]

Originally posted by Hurkyl
(except you should substitute back what u is)

Yeah I didn't feel like typing out any more latex