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Math Help - 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:

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

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

    \int csc^2\frac{\pi u}{4}\frac{du}{3}<br />

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

    Once you have \frac{1}{3}\int csc^2\frac{\pi u}{4}du make the further substitution v= \frac{\pi u}{4} or, equivalently, make the original substitution 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 u= \frac{\pi}{4}(3x+ 5), du= \frac{3\pi}{4}dx so dx= \frac{4}{3\pi}du. When x= 8/3, u= \frac{\pi}{4}(8+ 5)= \frac{13\pi}{4} and when x= 2, u= \frac{\pi}{4}(6+ 5)= \frac{11\pi}{4}. The integral becomes
    \frac{4}{3\pi}\int_{11\pi/4}^{13\pi/4} csc^2(u)du

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