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Math Help - Integration problem

  1. #1
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    Integration problem

    Evaluate the integral

    integral (0, pi/2) 60cos^5 xdx...

    I have no idea how to solve..
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  2. #2
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    Integration by parts will produce a reduction formula.
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  3. #3
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    Quote Originally Posted by Jgirl689 View Post
    Evaluate the integral

    integral (0, pi/2) 60cos^5 xdx...

    I have no idea how to solve..
    \int{\cos^5{x}\,dx} = \int{\cos^4{x}\cos{x}\,dx}

     = \int{(\cos^2{x})^2\cos{x}\,dx}

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

     = \int{(1 - 2\sin^2{x} + \sin^4{x})\cos{x}\,dx}.


    Now make the substitution u = \sin{x} so that \frac{du}{dx} = \cos{x}.
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  4. #4
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    many ways to solve this

    Quote Originally Posted by Jgirl689 View Post
    Evaluate the integral

    integral (0, pi/2) 60cos^5 xdx...

    I have no idea how to solve..
    i find several ways to solve this:
    1)you can expand cos^5x in terms of multiples of x using euler's formula considering y=cosx+isinx and 1/y=cosx-isinx.
    2)you can use a reduction formula as TKHunny said and
    3)the way 'prove it' did it
    4)and lastly u can perhaps use some property of definite integrals:
    integration(0,pi/2)f(x)=integration(0,pi/2)f(pi/2-x).

    i have not tested any of these methods though but i think they are all very feasible.
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  5. #5
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    Quote Originally Posted by Pulock2009 View Post
    i find several ways to solve this:
    1)you can expand cos^5x in terms of multiples of x using euler's formula considering y=cosx+isinx and 1/y=cosx-isinx.
    2)you can use a reduction formula as TKHunny said and
    3)the way 'prove it' did it
    4)and lastly u can perhaps use some property of definite integrals:
    integration(0,pi/2)f(x)=integration(0,pi/2)f(pi/2-x).

    i have not tested any of these methods though but i think they are all very feasible.
    Another alternative is to use trigonometric identities to convert it into functions such as \sin{nx} or \cos{nx}.
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