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

The integral from 2 to 6 of (2x+1)ln(x+1) is 42ln(7)-6ln(3)-16

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

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

42 ln (7) - 6 ln (3) - 16

Decimal

59.13655 \dots

Solution steps

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

Expand

(2 x + 1) ln (x + 1) : 2 x ln (x + 1) + ln (x + 1)

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

Apply the Sum Rule:

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

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

\int_{2}^{6} 2 x ln (x + 1) d x = - 12 + 35 ln (7) - 3 ln (3)

\int_{2}^{6} ln (x + 1) d x = 7 ln (7) - 4 - 3 ln (3)

= - 12 + 35 ln (7) - 3 ln (3) + 7 ln (7) - 4 - 3 ln (3)

Simplify

= 42 ln (7) - 6 ln (3) - 16

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What is the integral from 2 to 6 of (2x+1)ln(x+1) ?
The integral from 2 to 6 of (2x+1)ln(x+1) is 42ln(7)-6ln(3)-16