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Thread: Hyperbolic Function Integration

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    Hyperbolic Function Integration

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

    Would I change it to $\displaystyle e^x$. When I did, I got up to $\displaystyle \displaystyle\int \frac{2}{3e^x + e^{-x}} \, \mathrm{d}x$ 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 $\displaystyle \frac{\exp x}{\exp x}$ , it gives : $\displaystyle \frac{1}{3\exp x+\exp -x}=\frac{\exp x }{3(\exp x)^2+1}=\frac{\exp x }{(\sqrt{3}\exp x)^2+1}$

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

    Would I change it to $\displaystyle e^x$. When I did, I got up to $\displaystyle \displaystyle\int \frac{2}{3e^x + e^{-x}} \, \mathrm{d}x$ 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 :
    $\displaystyle t=e^x$

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

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


    Can you go on ? Multiply the bottom by t
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