1. ## 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.

2. 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 ?

3. Originally Posted by Air
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