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

The integral from 0 to 1 of x/(2x+1) is 1/4 (2-ln(3))

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

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

\frac{1}{4} (2 - ln (3))

Decimal

0.22534 \dots

Solution steps

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

Apply u-substitution

: \int_{1}^{3} \frac{u - 1}{4 u} d u

= \int_{1}^{3} \frac{u - 1}{4 u} d u

Take the constant out:

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

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

Expand

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

= \frac{1}{4} \cdot \int_{1}^{3} 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

= \frac{1}{4} (\int_{1}^{3} 1 d u - \int_{1}^{3} \frac{1}{u} d u)

\int_{1}^{3} 1 d u = 2

\int_{1}^{3} \frac{1}{u} d u = ln (3)

= \frac{1}{4} (2 - ln (3))

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What is the integral from 0 to 1 of x/(2x+1) ?
The integral from 0 to 1 of x/(2x+1) is 1/4 (2-ln(3))