What is the integral from-1 to 1 of (x^3)/(x+2) ?

The integral from-1 to 1 of (x^3)/(x+2) is 26/3-8ln(3)

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integral from-1 to 1 of (x^3)/(x+2)

\int_{- 1}^{1} \frac{x ^{3}}{x + 2} d x

\frac{26}{3} - 8 ln (3)

Decimal

- 0.12223 \dots

Solution steps

\int_{- 1}^{1} \frac{x ^{3}}{x + 2} d x

Apply u-substitution

= \int_{1}^{3} \frac{( u - 2 ) ^{3}}{u} d u

Expand

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

= \int_{1}^{3} 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_{1}^{3} u^{2} d u - \int_{1}^{3} 6 u d u + \int_{1}^{3} 12 d u - \int_{1}^{3} \frac{8}{u} d u

\int_{1}^{3} u^{2} d u = \frac{26}{3}

\int_{1}^{3} 6 u d u = 24

\int_{1}^{3} 12 d u = 24

\int_{1}^{3} \frac{8}{u} d u = 8 ln (3)

= \frac{26}{3} - 24 + 24 - 8 ln (3)

Simplify

= \frac{26}{3} - 8 ln (3)

What is the integral from-1 to 1 of (x^3)/(x+2) ?
The integral from-1 to 1 of (x^3)/(x+2) is 26/3-8ln(3)