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Thread: How to find this integral? guidance please.

  1. #1
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    How to find this integral? guidance please.

    Hi.

    I need to find the following integral:
    \int \sin^6x\cos^2xdx
    I just need a lead here.
    How should I approach this?
    I tried to use some trigonometric identities but couldn't get anywhere with it.
    If it indeed can be simplified with trigonometric identities, please point me to the right direction.
    Thanks in advanced!
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  2. #2
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    Re: How to find this integral? guidance please.

    Quote Originally Posted by Stormey View Post
    I need to find the following integral:
    \int \sin^6x\cos^2xdx

    Have a look at this.
    Thanks from topsquark
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    Re: How to find this integral? guidance please.

    How to find this integral? guidance please.-24-apr-13.png
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    Re: How to find this integral? guidance please.

    Thank you.
    that was very helpful!
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    Re: How to find this integral? guidance please.

    OK, this is not working out.
    this is what I've tried:

    \int \sin^6x\cos^3xdx
    t=\cos x
    so we get:
    \int \sin^6xt^3xdx
    and then:
    t^3=\cos^3x\Rightarrow t^6=\cos^6x\Rightarrow t^6=(\cos^2x)^3\Rightarrow t^6=(1-\sin^2x)^3
    now what?
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  6. #6
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    Re: How to find this integral? guidance please.

    Hang on, is your integral \displaystyle \begin{align*} \int{\sin^6{(x)}\cos^2{(x)}\,dx} \end{align*} like in the original post, or \displaystyle \begin{align*} \int{\sin^6{(x)}\cos^3{(x)}\,dx} \end{align*} like in this latest post?
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    Re: How to find this integral? guidance please.

    How to find this integral? guidance please.-25-apr-13.png
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    Re: How to find this integral? guidance please.

    OOPS! sorry...
    it's:
    \int \sin^6x\cos^2xdx

    and then:
    t=\sin x\Rightarrow t^2=\sin^2x\Rightarrow t^2=1-\cos^2x\Rightarrow \cos^2x=1-t^2

    so:
    \int t^6(1-t^2)dx

    how do I get rid of that dx?
    Last edited by Stormey; Apr 25th 2013 at 12:35 AM.
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  9. #9
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    Re: How to find this integral? guidance please.

    If you put t = sin x then it is cosx dx = dt
    Use the reduction formula Posted earlier
    Thanks from Stormey
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