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Thread: 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:

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

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

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

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

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

    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

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

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

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

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

    $\displaystyle u \int_1^2\frac1{(u-1)^2}\,du$ and $\displaystyle \frac1{(u-1)^2}\$ is $\displaystyle 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 $\displaystyle \frac{a}{u^2} + \frac{b}{u} + \frac{c}{u + 1}$
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  6. #6
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    Quote Originally Posted by wizzler View Post
    so $\displaystyle \int_1^2\frac1{u(u-1)^2}\,du$ can be put as

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

    We have

    $\displaystyle
    \begin{aligned}
    \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}
    \end{aligned}
    $

    Kill it now.
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