What is the integral from 0 to 1 of (2x)/(1+x) ?

The integral from 0 to 1 of (2x)/(1+x) is 2(1-ln(2))

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integral from 0 to 1 of (2x)/(1+x)

\int_{0}^{1} \frac{2 x}{1 + x} d x

2 (1 - ln (2))

Decimal

0.61370 \dots

Solution steps

\int_{0}^{1} \frac{2 x}{1 + x} d x

Take the constant out:

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

= 2 \cdot \int_{0}^{1} \frac{x}{1 + x} d x

Apply u-substitution

: \int_{1}^{2} \frac{u - 1}{u} d u

= 2 \cdot \int_{1}^{2} \frac{u - 1}{u} d u

Expand

\frac{u - 1}{u} : 1 - \frac{1}{u}

= 2 \cdot \int_{1}^{2} 1 - \frac{1}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int_{1}^{2} 1 d u - \int_{1}^{2} \frac{1}{u} d u)

\int_{1}^{2} 1 d u = 1

\int_{1}^{2} \frac{1}{u} d u = ln (2)

= 2 (1 - ln (2))

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What is the integral from 0 to 1 of (2x)/(1+x) ?
The integral from 0 to 1 of (2x)/(1+x) is 2(1-ln(2))