What is the integral from 0 to infinity of 77(e^x)/(e^{2x)+3} ?

The integral from 0 to infinity of 77(e^x)/(e^{2x)+3} is (77pi)/(3sqrt(3))

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integral from 0 to infinity of 77(e^x)/(e^{2x)+3}

\int_{0}^{\infty} 77 \frac{e ^{x}}{e ^{2 x} + 3} d x

\frac{77 π}{3 3}

Decimal Notation

46.55418 \dots

Solution steps

\int_{0}^{\infty} 77 \cdot \frac{e ^{x}}{e ^{2 x} + 3} d x

Compute the indefinite integral:

\int 77 \cdot \frac{e ^{x}}{e ^{2 x} + 3} d x = \frac{77}{3} arctan (\frac{1}{3} e^{x}) + C

Compute the boundaries:

\int_{0}^{\infty} 77 \cdot \frac{e ^{x}}{e ^{2 x} + 3} d x = \frac{77}{3} \cdot \frac{π}{2} - \frac{π}{6} \cdot \frac{77}{3}

= \frac{77}{3} \cdot \frac{π}{2} - \frac{π}{6} \cdot \frac{77}{3}

Simplify

= \frac{77 π}{3 3}

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What is the integral from 0 to infinity of 77(e^x)/(e^{2x)+3} ?
The integral from 0 to infinity of 77(e^x)/(e^{2x)+3} is (77pi)/(3sqrt(3))