What is the integral from 2 to 4 of (x^3)/(x+2) ?

The integral from 2 to 4 of (x^3)/(x+2) is -8ln(3/2)+44/3

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integral from 2 to 4 of (x^3)/(x+2)

\int_{2}^{4} \frac{x ^{3}}{x + 2} d x

- 8 ln (\frac{3}{2}) + \frac{44}{3}

Decimal Notation

11.42294 \dots

Solution steps

\int_{2}^{4} \frac{x ^{3}}{x + 2} d x

Apply u-substitution

: \int_{4}^{6} \frac{( u - 2 ) ^{3}}{u} d u

= \int_{4}^{6} \frac{( u - 2 ) ^{3}}{u} d u

Expand

\frac{( u - 2 ) ^{3}}{u} : u^{2} - 6 u + 12 - \frac{8}{u}

= \int_{4}^{6} u^{2} - 6 u + 12 - \frac{8}{u} d u

Apply the Sum Rule:

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

= \int_{4}^{6} u^{2} d u - \int_{4}^{6} 6 u d u + \int_{4}^{6} 12 d u - \int_{4}^{6} \frac{8}{u} d u

\int_{4}^{6} u^{2} d u = \frac{152}{3}

\int_{4}^{6} 6 u d u = 60

\int_{4}^{6} 12 d u = 24

\int_{4}^{6} \frac{8}{u} d u = 8 ln (\frac{3}{2})

= \frac{152}{3} - 60 + 24 - 8 ln (\frac{3}{2})

Simplify

\frac{152}{3} - 60 + 24 - 8 ln (\frac{3}{2}) : - 8 ln (\frac{3}{2}) + \frac{44}{3}

= - 8 ln (\frac{3}{2}) + \frac{44}{3}

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What is the integral from 2 to 4 of (x^3)/(x+2) ?
The integral from 2 to 4 of (x^3)/(x+2) is -8ln(3/2)+44/3