Just a hyperbolic integral

Mohamed Abdelkhaliq

New member
Joined
Dec 24, 2014
Messages
1
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,

Thanks in advance guys.
 
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,

Thanks in advance guys.

No guarantees but just maybe
u = \(\displaystyle \frac{1}{\sqrt{5 - e^{2x}}}\)
so that
e2x = 5 - u-2
Then use partial fractions
 
Or: start with \(\displaystyle u= e^{x}\) so that \(\displaystyle du= e^x dx\) or \(\displaystyle dx= \frac{1}{u}du\), \(\displaystyle \sqrt{5- e^{2x}}= \sqrt{5- u^2}\) and
\(\displaystyle \int \frac{dx}{\sqrt{5- e^{2x}}}= \int \frac{du}{u\sqrt{5- u^2}}\).

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