# Thread: Just a hyperbolic integral

1. ## Just a hyperbolic integral

I can't figure this one out,

integral of 1/(sqrt(5-e^(2x))

I don't see where I can make a substitution seeing as there is no e^x in the numerator,

2. Originally Posted by Mohamed Abdelkhaliq
I can't figure this one out,

integral of 1/(sqrt(5-e^(2x))

I don't see where I can make a substitution seeing as there is no e^x in the numerator,

No guarantees but just maybe
u = $\frac{1}{\sqrt{5 - e^{2x}}}$
so that
e2x = 5 - u-2
Then use partial fractions

3. Or: start with $u= e^{x}$ so that $du= e^x dx$ or $dx= \frac{1}{u}du$, $\sqrt{5- e^{2x}}= \sqrt{5- u^2}$ and
$\int \frac{dx}{\sqrt{5- e^{2x}}}= \int \frac{du}{u\sqrt{5- u^2}}$.

Now use a trig substitution: $u= \sqrt{5} sin(t)$ so that $du= \sqrt{5}cos(t)dt$, $\sqrt{5- u^2}= \sqrt{5(1- sin^2(t))}= \sqrt{5}cos(t)$ and
$\int \frac{du}{u\sqrt{5- u^2}}$$= \int \frac{\sqrt{5}cos(t)dt}{(\sqrt{5} sin(t))(\sqrt{5}cos(t))}=$$\frac{\sqrt{5}}{5}\int cosec(t)dt$