How to find the Integral of [e^(2x)/(4+e^(4X))]?

Question

I can not remember anything thing :(

Thank you in advance.

Answer 1

∫e^(2x)/(4+e^(4x)) dx

using the fact that e^(4x) = e^(2x) e^(2x)...

∫e^(2x)/(4+e^(2x)e^(2x)) dx

let u = e^(2x); du = 2 e^(2x)

1/2 ∫du /(4+u^2)

= 1/2 [1/4 arctan(u/4) + c]

= 1/8 arctan(e^(2x)/4) + c

Answer 2

∫e^2x/(4+e^4x)dx=

1/2∫1/(4+(e^2x)^2) d(e^2x)=

substitute e^2x=y obtain

1/2∫1/(4+y^2) dy=

1/4∫1/(1+(y/2)^2)d(y/2)

z=y/2

1/4∫1/(1+z^2) dz=

arctan z+C

substitute back yz to y and y to x