What is the integral from 5 to 3 of (x^2-36)/(x+6) ?

The integral from 5 to 3 of (x^2-36)/(x+6) is 4

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integral from 5 to 3 of (x^2-36)/(x+6)

\int_{5}^{3} \frac{x ^{2} - 36}{x + 6} d x

4

Solution steps

\int_{5}^{3} \frac{x ^{2} - 36}{x + 6} d x

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{3}^{5} \frac{x ^{2} - 36}{x + 6} d x

Expand

\frac{x ^{2} - 36}{x + 6} : \frac{x ^{2}}{x + 6} - \frac{36}{x + 6}

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 + 6} d x - \int_{3}^{5} \frac{36}{x + 6} d x)

\int_{3}^{5} \frac{x ^{2}}{x + 6} d x = 36 ln (11) - 4 - 72 ln (3)

\int_{3}^{5} \frac{36}{x + 6} d x = 36 (ln (11) - 2 ln (3))

= - (36 ln (11) - 4 - 72 ln (3) - 36 (ln (11) - 2 ln (3)))

Simplify

- (36 ln (11) - 4 - 72 ln (3) - 36 (ln (11) - 2 ln (3))) : 4

= 4

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What is the integral from 5 to 3 of (x^2-36)/(x+6) ?
The integral from 5 to 3 of (x^2-36)/(x+6) is 4