I tried integration by parts, but it doesn't look like I'm going to get down to somewhere that is integrable. Doesn't the e^2x/sqrt(1+e^2x) just get messier and messier? Thanks!
For the integral part, you can use substitution. u-subbing is mostly used.
For this you could let \(\displaystyle \L\\u=\sqrt{1+e^{2x}}, \;\ du=\frac{e^{2x}}{\sqrt{1+e^{2x}}}dx, \;\ \sqrt{u^{2}-1}=e^{x}\)
This gives:
\(\displaystyle \L\\\int\sqrt{u^{2}-1}du\)
For this, you could use trig sub, u=sec(θ),du=sec(θ)tan(θ)dθ
Or some other substitution. Try working through this. It's good practice and maybe you'll get a sense of accomplishment when you're done.
This will require Integration By Parts, unless you already know the formula: . . . . \(\displaystyle \L\frac{1}{2}\left[\sec\theta\tan\theta \,+\,\ln\left|\sec\theta\,+\,\tan\theta\right|\right]\)
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