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

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

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

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

- \frac{1}{e ^{x} + 1} + C

Solution steps

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

Factor

\frac{e ^{x}}{1 + 2 e ^{x} + e ^{2 x}}

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

Apply u-substitution

: \int \frac{1}{u ^{2}} d u

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

Apply the Power Rule

: - \frac{1}{u}

= - \frac{1}{u}

Substitute back

u = e^{x} + 1

= - \frac{1}{e ^{x} + 1}

Add a constant to the solution

= - \frac{1}{e ^{x} + 1} + C

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