What is the integral from 4 to 6 of (x+1)ln(2x+4) ?

The integral from 4 to 6 of (x+1)ln(2x+4) is ln((2^{72})/(531441))-5

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integral from 4 to 6 of (x+1)ln(2x+4)

\int_{4}^{6} (x + 1) ln (2 x + 4) d x

ln (\frac{2 ^{72}}{531441}) - 5

Decimal

31.72324 \dots

Solution steps

\int_{4}^{6} (x + 1) ln (2 x + 4) d x

Apply Integration By Parts

: [ln (2 x + 4) (\frac{x ^{2}}{2} + x) - \int \frac{x}{2} d x]_{4}^{6}

= [ln (2 x + 4) (\frac{x ^{2}}{2} + x) - \int \frac{x}{2} d x]_{4}^{6}

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

= [ln (2 x + 4) (\frac{x ^{2}}{2} + x) - \frac{x ^{2}}{4}]_{4}^{6}

Compute the boundaries:

96 ln (2) - 5 - 12 ln (12)

= 96 ln (2) - 5 - 12 ln (12)

Simplify

= ln (\frac{2 ^{72}}{531441}) - 5

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What is the integral from 4 to 6 of (x+1)ln(2x+4) ?
The integral from 4 to 6 of (x+1)ln(2x+4) is ln((2^{72})/(531441))-5