What is the integral from 3 to 5 of (x^2+x+6)/(x-2) ?

The integral from 3 to 5 of (x^2+x+6)/(x-2) is 12ln(3)+14

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integral from 3 to 5 of (x^2+x+6)/(x-2)

\int_{3}^{5} \frac{x ^{2} + x + 6}{x - 2} d x

12 ln (3) + 14

Decimal

27.18334 \dots

Solution steps

\int_{3}^{5} \frac{x ^{2} + x + 6}{x - 2} d x

Expand

\frac{x ^{2} + x + 6}{x - 2} : \frac{x ^{2}}{x - 2} + \frac{x}{x - 2} + \frac{6}{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_{3}^{5} \frac{x ^{2}}{x - 2} d x + \int_{3}^{5} \frac{x}{x - 2} d x + \int_{3}^{5} \frac{6}{x - 2} d x

\int_{3}^{5} \frac{x ^{2}}{x - 2} d x = 4 ln (3) + 12

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

\int_{3}^{5} \frac{6}{x - 2} d x = 6 ln (3)

= 4 ln (3) + 12 + 2 + 2 ln (3) + 6 ln (3)

Simplify

= 12 ln (3) + 14

What is the integral from 3 to 5 of (x^2+x+6)/(x-2) ?
The integral from 3 to 5 of (x^2+x+6)/(x-2) is 12ln(3)+14