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

The integral from 0 to 1 of xln(2x+1) is 3/8 ln(3)

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

\int_{0}^{1} x ln (2 x + 1) d x

\frac{3}{8} ln (3)

Decimal Notation

0.41197 \dots

Solution steps

\int_{0}^{1} x ln (2 x + 1) d x

Apply u-substitution

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

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

Apply Integration By Parts

: [ln (u) (\frac{u ^{2}}{2} - u) - \int \frac{1}{2} (u - 2) d u]_{1}^{3}

= \frac{1}{4} [ln (u) (\frac{u ^{2}}{2} - u) - \int \frac{1}{2} (u - 2) d u]_{1}^{3}

\int \frac{1}{2} (u - 2) d u = \frac{u ^{2}}{4} - u

= \frac{1}{4} [ln (u) (\frac{u ^{2}}{2} - u) - (\frac{u ^{2}}{4} - u)]_{1}^{3}

Simplify

\frac{1}{4} [ln (u) (\frac{u ^{2}}{2} - u) - (\frac{u ^{2}}{4} - u)]_{1}^{3} : \frac{1}{4} [\frac{1}{2} u^{2} ln (u) - u ln (u) - \frac{u ^{2}}{4} + u]_{1}^{3}

= \frac{1}{4} [\frac{1}{2} u^{2} ln (u) - u ln (u) - \frac{u ^{2}}{4} + u]_{1}^{3}

Compute the boundaries:

\frac{3}{2} ln (3)

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

Simplify

= \frac{3}{8} ln (3)

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What is the integral from 0 to 1 of xln(2x+1) ?
The integral from 0 to 1 of xln(2x+1) is 3/8 ln(3)