What is the integral of ((e^{2x}))/(1+e^{4x)} ?

The integral of ((e^{2x}))/(1+e^{4x)} is 1/2 arctan(e^{2x})+C

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integral of ((e^{2x}))/(1+e^{4x)}

\int \frac{( e ^{2 x} )}{1 + e ^{4 x}} d x

\frac{1}{2} arctan (e^{2 x}) + C

Solution steps

\int \frac{e ^{2 x}}{1 + e ^{4 x}} d x

Apply u-substitution

: \int \frac{1}{2 ( 1 + u ^{2} )} d u

= \int \frac{1}{2 ( 1 + u ^{2} )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{1 + u ^{2}} d u

Use the common integral:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= \frac{1}{2} arctan (u)

Substitute back

u = e^{2 x}

= \frac{1}{2} arctan (e^{2 x})

Add a constant to the solution

= \frac{1}{2} arctan (e^{2 x}) + C

\int 6 sin^{3} (4 x) cos^{3} (4 x) d x

\frac{d y}{d x} = \frac{x ^{3} - 2 y}{x}

\int \frac{( 3 x + 2 )}{19 - 5 x + x ^{2}} d x

x^{3} - 4 x^{2}

y^{^{'}} = - a y^{2} + b

What is the integral of ((e^{2x}))/(1+e^{4x)} ?
The integral of ((e^{2x}))/(1+e^{4x)} is 1/2 arctan(e^{2x})+C