What is antiderivative of y=e^2x/e^4x+1?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You may find the antiderivative of a function evaluating the indefinite integral of the function such that:

`int (e^2x)/(e^(4x)+1) dx`

You should use substitution method such that:

`e^(4x) = (e^(2x))^2 = y^2 => y =e^(2x) => dy = 2e^(2x)dx => e^(2x)dx = (dy)/2`

Changing the variable of integrand yields:

`int (e^2x)/(e^(4x)+1) dx = int ((dy)/2)/(y^2 + 1)`

`int ((dy)/2)/(y^2 + 1) = (1/2)int (dy)/(y^2 + 1)`

`int ((dy)/2)/(y^2 + 1) = (1/2)tan^(-1)(y) + c`

`int (e^2x)/(e^(4x)+1) dx = (1/2)tan^(-1)(e^(2x)) + c`

Hence, evaluating the antiderivative of the given function yields `int (e^2x)/(e^(4x)+1) dx = (1/2)tan^(-1)(e^(2x)) + c.`

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