What could I use as a u-sub?

Question

luigi0210 · Answer

$$\LARGE \int~\frac{e^x}{3-e^{2x}}~dx$$

muzzack · Answer

The method of u-substitution is a method for algebraically simplifying the form of a function so that its antiderivative can be easily recognized. This method is intimately related to the chain rule for differentiation. For example, since the derivative of $$e ^{x}$$ is $$D(e ^{x}) = e ^{x}$$
it follows easily that $$\int\limits(e ^{x})dx=e ^{x}+C$$

muzzack · Answer

@mathmale

anonymous · Answer

$$u=3-e^{2x}~~\Rightarrow~~du=-2e^{2x}~dx~~\iff~~-\frac{1}{2}du=e^{2x}~dx$$
$$\int\frac{e^{2x}}{3-e^{2x}}~dx=-\frac{1}{2}\int\frac{du}{u}$$

anonymous · Answer

Oh sorry, I thought there was an $e^{2x}$ in the numerator...

luigi0210 · Answer

Yea, that's my problem.

anonymous · Answer

A trig sub should work: $e^x=\sqrt 3\sin u$, so that $e^x~dx=\sqrt3\cos u~du~~\iff~~dx=\dfrac{\sqrt3\cos u}{\sqrt3\sin u}~du=\cot u~du$

luigi0210 · Answer

Alright, thank you siths and muzzack :)

anonymous · Answer

yw