Question

Evaluate the integral: the integral of [5e^(2t)]/[1+4e^(2t)]dt. I used u sub and let u=e^2t and got 5/2arctan(e^2t)+C. But this answer is incorrect. Please help. Thanks

Answer 1

There are certain pattern that you should recognize in derivatives

e.g. if y = ln(anything)
then dy/dx = derivative(anything)/anything

I noticed that the derivative of the denominator appears at the top, differing only in a constant

so let's start with
y = ln(1 + 4e^(2t))
dy/dt = 8e^(2t)/(1 + 4e^(2t) )

instead of the 5 I need on top I see an 8

so how about

y = (5/8)ln(1 + 4e^(2t) ) + C