What is the integral from 0 to 5 of (3x+11)/(x+2) ?

The integral from 0 to 5 of (3x+11)/(x+2) is -5ln(2)+5ln(7)+15

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integral from 0 to 5 of (3x+11)/(x+2)

\int_{0}^{5} \frac{3 x + 11}{x + 2} d x

- 5 ln (2) + 5 ln (7) + 15

Decimal

21.26381 \dots

Solution steps

\int_{0}^{5} \frac{3 x + 11}{x + 2} d x

Expand

\frac{3 x + 11}{x + 2} : \frac{3 x}{x + 2} + \frac{11}{x + 2}

Apply the Sum Rule:

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

= \int_{0}^{5} \frac{3 x}{x + 2} d x + \int_{0}^{5} \frac{11}{x + 2} d x

\int_{0}^{5} \frac{3 x}{x + 2} d x = 3 (5 - 2 (ln (7) - ln (2)))

\int_{0}^{5} \frac{11}{x + 2} d x = 11 (ln (7) - ln (2))

= 3 (5 - 2 (ln (7) - ln (2))) + 11 (ln (7) - ln (2))

Simplify

3 (5 - 2 (ln (7) - ln (2))) + 11 (ln (7) - ln (2)) : - 5 ln (2) + 5 ln (7) + 15

= - 5 ln (2) + 5 ln (7) + 15

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What is the integral from 0 to 5 of (3x+11)/(x+2) ?
The integral from 0 to 5 of (3x+11)/(x+2) is -5ln(2)+5ln(7)+15