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Math Help - Good examples on Integrals

  1. #1
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    Good examples on Integrals

    Hey!

    Can anyone give me good examples on how to solve integrals like:

    \int_{0}^{\pi/2} \frac{cos x}{sin^2x + sin^3 x} dx

    with partial integration/substitution methods?
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  2. #2
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    First rewrite the integrand as follows

    \int_0^{\pi/2}\frac{\cos x}{\sin^2x+\sin^3x}\,dx=\int_0^{\pi/2}\frac{\cos x}{\sin^2x(1+\sin x)}\,dx

    Then set u=1+\sin x\implies du=\cos x\,dx, which yields

    \int_1^2\frac1{u(u-1)^2}\,du

    Just post if you cannot take it from there.
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  3. #3
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    Just be careful this is a "Improper Integral of the 2nd Type".
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  4. #4
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    Quote Originally Posted by Krizalid View Post
    First rewrite the integrand as follows

    \int_0^{\pi/2}\frac{\cos x}{\sin^2x+\sin^3x}\,dx=\int_0^{\pi/2}\frac{\cos x}{\sin^2x(1+\sin x)}\,dx

    Then set u=1+\sin x\implies du=\cos x\,dx, which yields

    \int_1^2\frac1{u(u-1)^2}\,du

    Just post if you cannot take it from there.
    so \int_1^2\frac1{u(u-1)^2}\,du can be put as

    u \int_1^2\frac1{(u-1)^2}\,du and \frac1{(u-1)^2}\ is ln((u-1)^2) or am i thinking it totally wrong now?
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  5. #5
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    neuu of course...

    i part integrate it so i get \frac{a}{u^2} + \frac{b}{u} + \frac{c}{u + 1}
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  6. #6
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    Quote Originally Posted by wizzler View Post
    so \int_1^2\frac1{u(u-1)^2}\,du can be put as

    u \int_1^2\frac1{(u-1)^2}\,du
    This step is not correct 'cause "u" it's a function of "x", not a constant.

    We have

     <br />
\begin{aligned}<br />
\frac1{u(u-1)^2}&=\frac{u^2-2u+1+u-u^2+u}{u(u-1)^2}\\&=\frac{(u-1)^2+u-u(u-1)}{u(u-1)^2}\\&=\frac1u+\frac1{(u-1)^2}-\frac1{u-1}<br />
\end{aligned}<br />

    Kill it now.
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