What is the integral from 0 to ln(sqrt(3 of))(4e^x)/(1+e^{2x)} ?

The integral from 0 to ln(sqrt(3 of))(4e^x)/(1+e^{2x)} is pi/3

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integral from 0 to ln(sqrt(3 of))(4e^x)/(1+e^{2x)}

\int_{0}^{l n (3)} \frac{4 e ^{x}}{1 + e ^{2 x}} d x

\frac{π}{3}

Decimal

1.04719 \dots

Solution steps

\int_{0}^{l n (3)} \frac{4 e ^{x}}{1 + e ^{2 x}} d x

Take the constant out:

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

= 4 \cdot \int_{0}^{l n (3)} \frac{e ^{x}}{1 + e ^{2 x}} d x

Apply u-substitution

= 4 \cdot \int_{1}^{3} \frac{1}{1 + u ^{2}} d u

Use the common integral:

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

= 4 [arctan (u)]_{1}^{3}

Compute the boundaries:

\frac{π}{12}

= 4 \cdot \frac{π}{12}

Simplify

= \frac{π}{3}

What is the integral from 0 to ln(sqrt(3 of))(4e^x)/(1+e^{2x)} ?
The integral from 0 to ln(sqrt(3 of))(4e^x)/(1+e^{2x)} is pi/3