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

The integral from-5 to 2 of x/((x+6)^{2/3)} is -27/4

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

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

- \frac{27}{4}

Decimal

- 6.75

Solution steps

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

Apply u-substitution

: \int_{1}^{8} \frac{u - 6}{u ^{\frac{2}{3}}} d u

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

Expand

\frac{u - 6}{u ^{\frac{2}{3}}} : u^{\frac{1}{3}} - \frac{6}{u ^{\frac{2}{3}}}

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

\int_{1}^{8} u^{\frac{1}{3}} d u = \frac{45}{4}

\int_{1}^{8} \frac{6}{u ^{\frac{2}{3}}} d u = 18

= \frac{45}{4} - 18

Simplify

\frac{45}{4} - 18 : - \frac{27}{4}

= - \frac{27}{4}

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What is the integral from-5 to 2 of x/((x+6)^{2/3)} ?
The integral from-5 to 2 of x/((x+6)^{2/3)} is -27/4