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Thread: integration of product of two dissimlar expressions

  1. #1
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    integration of product of two dissimlar expressions

    I have to integrate by substitution:

    $\displaystyle \int csc^2(3x+5)\frac{\pi}{4}dx
    $ between 8/3 and 2

    tried making (3x+5)=u to give:

    $\displaystyle \int csc^2\frac{\pi u}{4}\frac{du}{3}
    $

    I'm not sure if this is correct though, and I don't know how to integrate trig functions when the variable has a coefficient. Please help.
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  2. #2
    MHF Contributor

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    So the $\displaystyle \frac{\pi}{4}$ is inside the csc? That is, it is $\displaystyle csc^2(\frac{\pi}{4}(3x+5))$ which is not quite what you wrote.

    Once you have $\displaystyle \frac{1}{3}\int csc^2\frac{\pi u}{4}du$ make the further substitution $\displaystyle v= \frac{\pi u}{4}$ or, equivalently, make the original substitution $\displaystyle u= \frac{\pi}{4}(3x+ 5)$ rather than just u= 3x+ 5.
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  3. #3
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    Hi, the pi/4 is within the csc, yes. The answer is supposed to be 0 but i just cant seem to get that.
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  4. #4
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    With the substitution $\displaystyle u= \frac{\pi}{4}(3x+ 5)$, $\displaystyle du= \frac{3\pi}{4}dx$ so $\displaystyle dx= \frac{4}{3\pi}du$. When x= 8/3, $\displaystyle u= \frac{\pi}{4}(8+ 5)= \frac{13\pi}{4}$ and when x= 2, $\displaystyle u= \frac{\pi}{4}(6+ 5)= \frac{11\pi}{4}$. The integral becomes
    $\displaystyle \frac{4}{3\pi}\int_{11\pi/4}^{13\pi/4} csc^2(u)du$

    Now, the anti-derivative of $\displaystyle csc^2(u)$ is $\displaystyle - cot(u)$. Evaluated at $\displaystyle \frac{13\pi}{4}$ that is 1 and at $\displaystyle \frac{11\pi}{4}$ it is -1. The value of the integral is NOT 0, it is $\displaystyle \frac{8}{3\pi}$.
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