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Math Help - Integrating

  1. #1
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    Integrating

    hi, I was wondering if someone could show me how to do a problem. I'm studying for a test...and I wanted to be prepared in case something like this came up.

    how to you integrate the following...

    cosine cubed of x * sin to the fifth of x dx
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  2. #2
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    Quote Originally Posted by clockingly View Post
    hi, I was wondering if someone could show me how to do a problem. I'm studying for a test...and I wanted to be prepared in case something like this came up.

    how to you integrate the following...

    cosine cubed of x * sin to the fifth of x dx
    This happens to be a long integral.

    But the first set is to use integration by part,
    u=x and v'=sin^5 x

    Thus, u'=1
    But when you find v you need to find the integral of sin^5 x which you can do by writing it as:
    sin^5 x=sin^4 x sin x = (1-cos^2x)^2*sin x
    Then use the substitution t=sin x.

    But after you do that long integration by parts you will need to do another trigonometric integral like the one above.
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  3. #3
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    Hello, clockingly!

    cos^3(x)·sin^5(x) dx
    We have: .cos^2(x)·sin^5(x) · cos(x)

    Then: .[1 - sin^2(x)]·sin^5(x) · cos(x)

    And: .[sin^5(x) - sin^7(x)]·cos(x)


    We can integrate: .
    [sin^5(x) - sin^7(x)]·cos(x) dx

    . . with the substitution: .u = sin(x)

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