What is the integral from 1 to 2 of (3x^2+2)ln(2x) ?

The integral from 1 to 2 of (3x^2+2)ln(2x) is 21ln(2)-13/3

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integral from 1 to 2 of (3x^2+2)ln(2x)

\int_{1}^{2} (3 x^{2} + 2) ln (2 x) d x

21 ln (2) - \frac{13}{3}

Decimal

10.22275 \dots

Solution steps

\int_{1}^{2} (3 x^{2} + 2) ln (2 x) d x

Apply Integration By Parts

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

= [ln (2 x) (x^{3} + 2 x) - \int x^{2} + 2 d x]_{1}^{2}

\int x^{2} + 2 d x = \frac{x ^{3}}{3} + 2 x

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

Expand

ln (2 x) (x^{3} + 2 x) - (\frac{x ^{3}}{3} + 2 x) : x^{3} ln (2 x) + 2 x ln (2 x) - \frac{x ^{3}}{3} - 2 x

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

Compute the boundaries:

21 ln (2) - \frac{13}{3}

= 21 ln (2) - \frac{13}{3}

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What is the integral from 1 to 2 of (3x^2+2)ln(2x) ?
The integral from 1 to 2 of (3x^2+2)ln(2x) is 21ln(2)-13/3