# int e^(3x)/(e^x+e^(3x)) dx

int e^(3x)/(e^x+e^(3x))dx

To solve this, let's simplify first the integrand.

=int e^(3x)/(e^x(1+e^(2x)))dx

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

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

Then, apply u-substitution method.

u=1+e^(2x)

du = e^(2x)*2dx

(du)/2=e^(2x)dx

Expressing the integral in terms of u, it becomes:

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

= int 1/u * (du)/2

= 1/2...

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int e^(3x)/(e^x+e^(3x))dx

To solve this, let's simplify first the integrand.

=int e^(3x)/(e^x(1+e^(2x)))dx

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

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

Then, apply u-substitution method.

u=1+e^(2x)

du = e^(2x)*2dx

(du)/2=e^(2x)dx

Expressing the integral in terms of u, it becomes:

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

= int 1/u * (du)/2

= 1/2 int 1/u du

=1/2ln|u|+ C

And, substitute back u = 1+e^(2x) .

=1/2ln|1+e^(2x)|+C

Therefore, int e^(3x)/(e^x+e^(3x))dx = 1/2ln|1+e^(2x)| + C` .

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