How do you find the antiderivative of $(e^(2x))/(1+(e^(4x))dx$ ?

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Answer 1 · 2016-12-12T20:46:54

$1/2arctan(e^(2x))+C$

$inte^(2x)/(1+e^(4x))dx$

Let $u=e^(2x)$ so $du=2e^(2x)dx$ .

$=1/2int(2e^(2x))/(1+(e^(2x))^2)dx$

$=1/2int1/(1+u^2)du$

This is the arctangent integral:

$=1/2arctan(u)+C$

$=1/2arctan(e^(2x))+C$

Another way to show this is to use the trigonometric substitution $e^(2x)=tan(theta)$ .

How do you find the antiderivative of (e^(2x))/(1+(e^(4x))dx(e^(2x))/(1+(e^(4x))dx?