Trigonometric Integral

1. Feb 23, 2013

whatlifeforme

1. The problem statement, all variables and given/known data
solve the integral.

2. Relevant equations
integral (5pi/6 to pi) (cosx)^4 / sqrt(1-sinx) dx

3. The attempt at a solution
Following the solution manual:

integral (5pi/6 to pi) ( (cosx)^4 / sqrt(1-sinx)) * sqrt(1+sinx)/sqrt(1+sinx)

however i am not sure how to do the algebra here.

it should convert to integral (5pi/6 to pi) ( (cosx)^4 / sqrt(1-sin^2x)) * sqrt(1+sinx) dx

2. Feb 23, 2013

LCKurtz

When you say "it should convert" do you mean you don't see how do get that last step or you don't know what to do next? You do know that $1-\sin^2 x = \cos^2 x$, right? You could change all the even powers of $\cos x$ to sines and do a u substitution.

3. Feb 23, 2013

whatlifeforme

solved.

interal (sin2x)^4 / (sqrt(1-cos2x) dx

4. Feb 23, 2013

LCKurtz

Those problems are so similar that whatever you did for the first one (which you haven't shown us) should work on the second one.

5. Feb 23, 2013

whatlifeforme

i can't remember what i did and i lost the scrap piece of paper. i think i multiplied by the conjugate but im stuck.

6. Feb 23, 2013

SammyS

Staff Emeritus
That should work.

What are you stuck with?

7. Feb 23, 2013

whatlifeforme

after multiplying by the conjugate and simplifying i get: integral (sin2x)^3 * sqrt(1+cos2x) dx

8. Feb 23, 2013

SammyS

Staff Emeritus
OK. So you have
$\displaystyle \int (\sin(2x))^3\sqrt{1+\cos(2x)\,}\,dx\ .$​

What did LCKurtz have you do in post #2 ?
This time all the arguments are 2x, rather than x, but so what?

9. Feb 23, 2013

whatlifeforme

there are no even power cosines.

10. Feb 23, 2013

iRaid

Got it.
Take u=2x and then you have to use the fact that sin^3x=sin^2x*sinx...

Last edited: Feb 23, 2013
11. Feb 24, 2013

SammyS

Staff Emeritus
There certainly is an even power of cosine.

cos3(2x)=cos2(2x)∙cos(2x).

There's a cos2 for you.

12. Feb 24, 2013

whatlifeforme

what about the square root with 1+cos2x, though?

the problem is (sin2x)^3 not (cos2x)^3.

13. Feb 24, 2013

whatlifeforme

in my solutions manual, in has the following as part of the solution. how is this correct algebra?

(-2/3) + (2/3)(3/2)^(3/2) --> = sqrt(3/2) - 2/3

what happened to the 3 in the exponent (3/2)??

14. Feb 24, 2013

Karnage1993

(2/3) is the same as 1/(3/2) = (3/2)^-1.

15. Feb 24, 2013

SammyS

Staff Emeritus
My mistake !

(sin2x)^3 = (sin2x)2(sin2x)

= (1-cos2(2x))sin(2x)

That all suggests to me the substitution, u = cos(2x) .

Perhaps better yet is u = 1 + cos(2x). Do you see why?

16. Feb 24, 2013

LCKurtz

@whatlifeforme: What I don't get about this thread is that in post #3 you claimed you have solved your original problem. Then you posted an almost identical problem just changing the sines to cosines and the x's to 2x's. So why are you going on and on with questions about every little step when you have already solved an essentially identical problem? What's going on here??

17. Feb 24, 2013

iRaid

Then you do another u substitution and then an integration by parts from what I remember doing.