`int` `e^(3x)/(e^x+e^(3x)) dx`
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Lix Lemjay
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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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