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

1 Answer
Jun 21, 2016

Answer:

# 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.