What is the integral from 1 to 5 of (2x+1)ln(x+1) ?

The integral from 1 to 5 of (2x+1)ln(x+1) is 24ln(6)+2ln(108)-12

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integral from 1 to 5 of (2x+1)ln(x+1)

\int_{1}^{5} (2 x + 1) ln (x + 1) d x

24 ln (6) + 2 ln (108) - 12

Decimal

40.36648 \dots

Solution steps

\int_{1}^{5} (2 x + 1) ln (x + 1) d x

Expand

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

= \int_{1}^{5} 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_{1}^{5} 2 x ln (x + 1) d x + \int_{1}^{5} ln (x + 1) d x

\int_{1}^{5} 2 x ln (x + 1) d x = 2 (12 ln (6) - 4)

\int_{1}^{5} ln (x + 1) d x = 2 ln (108) - 4

= 2 (12 ln (6) - 4) + 2 ln (108) - 4

Simplify

2 (12 ln (6) - 4) + 2 ln (108) - 4 : 24 ln (6) + 2 ln (108) - 12

= 24 ln (6) + 2 ln (108) - 12

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What is the integral from 1 to 5 of (2x+1)ln(x+1) ?
The integral from 1 to 5 of (2x+1)ln(x+1) is 24ln(6)+2ln(108)-12