Find the Indefinite Integral Using Substitution: e^2x and (1+e^2x)^3

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Homework Statement


Find the indefinite integral using substitution:
\inte^2x(1+e^2x)^3 dx

Homework Equations





The Attempt at a Solution


I'm not sure how to start. What do I substitute? Any suggestions to get me started will be greatly appreciated.
 
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put u=e^2x====>du=2e^2xdx
 
Better to put u=(1+e^(2x)).
 
Dick said:
Better to put u=(1+e^(2x)).

Yes, thanks dick. Thats what I ended up doing. I had just learned this stuff today, so I think msa's approach may have been a bit over my head.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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