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Math Help - [SOLVED] Trigonometric Integral

  1. #1
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    [SOLVED] Trigonometric Integral

    How is this integrated?

    \int{\cos^2{\Theta}}d\Theta
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  2. #2
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    Quote Originally Posted by Mike9182 View Post
    How is this integrated?

    \int{\cos^2{\Theta}}d\Theta
    Racall the identity: cos^2(\theta)=\frac{1}{2}(cos(2\theta)+1).
    Hence:
    \int cos^2(\theta) d\theta = \frac{1}{2} \int cos(2\theta) d\theta + \frac{1}{2} \int d\theta .

    It should be easy now.
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  3. #3
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    How do you integrate \frac{1}{2} \int(\cos{2\Theta})d\Theta?
    Last edited by Mike9182; January 26th 2010 at 04:18 PM. Reason: Latex error
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    Quote Originally Posted by Mike9182 View Post
    How do you integrate \frac{1}{2} \int(\cos{2\Theta})d\Theta?

     <br />
\int(\cos{k\Theta})d\Theta, k\in \mathbb{R} = \frac{1}{k} \sin{k\Theta}+C<br />
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  5. #5
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    Quote Originally Posted by Mike9182 View Post
    How do you integrate \frac{1}{2} \int(\cos{2\Theta})d\Theta?
    there is two ways:
    1- Substitute u=2\theta.
    2- Apply this formula:
    b\int cos(a\theta) d\theta = \frac{b}{a} sin(a\theta) + C for a \neq 0
    in your question: a=2 and b=\frac{1}{2}.

    Quote Originally Posted by pickslides View Post
     <br />
\int(\cos{k\Theta})d\Theta, k\in \mathbb{R} = \frac{1}{k} \sin{k\Theta}+C<br />
    k \neq 0
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  6. #6
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    Thanks, I understand now.
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  7. #7
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    Quote Originally Posted by Mike9182 View Post
    How is this integrated?

    \int{\cos^2{\Theta}}d\Theta
    Another way...

    \int{Cos\theta\ Cos\theta}d\theta=\int{u}dv=uv-\int{v}du

    u=Cos\theta,\ \frac{du}{d\theta}=-Sin\theta,\ du=-Sin\theta\ d\theta

    dv=Cos\theta\ d\theta,\ v=\int{Cos\theta}d\theta=Sin\theta

    uv-\int{v}du=Cos\theta Sin\theta+\int{Sin^2\theta}d\theta

    \Rightarrow\ I=Cos\theta Sin\theta+\int{(1-Cos^2\theta)}d\theta=Cos\theta Sin\theta+\int{(1)}d\theta-I

    2I=Cos\theta Sin\theta+\theta\ \Rightarrow\ I=\frac{1}{2}Cos\theta Sin\theta+\frac{\theta}{2}+C=\frac{1}{4}Sin2\theta  +\frac{\theta}{2}+C
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