Question

How do you find the antiderivative of `e^(2x) / (3+e^(2x))`?

Answer 1

`int e^(2x) / (3+e^(2x)) \ dx = ln sqrt{ 3 + e^{2x)) + C`

Explanation:

easiest way always is to recognise the patterns

generalisation `d/(dx) ln (f(x)) = ( f'(x) ) / f(x)`

so if we consider `d/(dx) ln (3 + e^{2x}) = 1/(3 + e^{2x}) * 2 e^{2x}` then we're pretty much done

because `d/(dx) ln (3 + e^{2x}) = ( 2 e^{2x})/(3 + e^{2x})` then we actually want `1/2 * d/(dx) ln (3 + e^{2x}) = d/(dx) [ 1/2 * ln (3 + e^{2x})]` moving the constant inside the derivative

`= d/(dx) [ ln sqrt{ 3 + e^{2x)) \ ]`

thusly

`int e^(2x) / (3+e^(2x)) \ dx = ln sqrt{ 3 + e^{2x)) + C`

you can plough through a whole series of subs but seeing the pattern is a real life saver.