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Math Help - Hyperbolic Function Integration

  1. #1
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    Hyperbolic Function Integration

    Q: \displaystyle\int \frac{1}{\mathrm{sinh}x + 2 \mathrm{cosh}x} \, \mathrm{d}x

    Would I change it to e^x. When I did, I got up to \displaystyle\int \frac{2}{3e^x + e^{-x}} \, \mathrm{d}x but didn't know how to proceed further. Can someone show me how you would have done the original question. Thanks in advance.
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    Super Member flyingsquirrel's Avatar
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    Hello

    If one multiplies by \frac{\exp x}{\exp x} , it gives : \frac{1}{3\exp x+\exp -x}=\frac{\exp x }{3(\exp x)^2+1}=\frac{\exp x }{(\sqrt{3}\exp x)^2+1}

    Do you see the derivative of x\mapsto \arctan x which is appearing ?
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  3. #3
    Moo
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    Quote Originally Posted by Air View Post
    Q: \displaystyle\int \frac{1}{\mathrm{sinh}x + 2 \mathrm{cosh}x} \, \mathrm{d}x

    Would I change it to e^x. When I did, I got up to \displaystyle\int \frac{2}{3e^x + e^{-x}} \, \mathrm{d}x but didn't know how to proceed further. Can someone show me how you would have done the original question. Thanks in advance.

    Right transformation ^^

    Now, substitute :
    t=e^x

    \frac{dt}{dx}=e^x=t --> dx=\frac{dt}{t}

    ---> \int \frac{2}{3e^x + e^{-x}} dx=\int \frac{2}{3t+\frac 1t} \cdot \frac 1t dt


    Can you go on ? Multiply the bottom by t
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