What is the integral from 2 to 6 of x^2 1/8 (x-2) ?

The integral from 2 to 6 of x^2 1/8 (x-2) is 68/3

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integral from 2 to 6 of x^2 1/8 (x-2)

\int_{2}^{6} x^{2} \frac{1}{8} (x - 2) d x

\frac{68}{3}

Decimal

22.66666 \dots

Solution steps

\int_{2}^{6} x^{2} \frac{1}{8} (x - 2) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{8} \cdot \int_{2}^{6} x^{2} (x - 2) d x

Expand

x^{2} (x - 2) : x^{3} - 2 x^{2}

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

Apply the Sum Rule:

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

= \frac{1}{8} (\int_{2}^{6} x^{3} d x - \int_{2}^{6} 2 x^{2} d x)

\int_{2}^{6} x^{3} d x = 320

\int_{2}^{6} 2 x^{2} d x = \frac{416}{3}

= \frac{1}{8} (320 - \frac{416}{3})

Simplify

\frac{1}{8} (320 - \frac{416}{3}) : \frac{68}{3}

= \frac{68}{3}

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What is the integral from 2 to 6 of x^2 1/8 (x-2) ?
The integral from 2 to 6 of x^2 1/8 (x-2) is 68/3