What is the integral from 0 to 1 of-3x(-2x^2-1)^3 ?

The integral from 0 to 1 of-3x(-2x^2-1)^3 is 15

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integral from 0 to 1 of-3x(-2x^2-1)^3

\int_{0}^{1} - 3 x (- 2 x^{2} - 1)^{3} d x

15

Solution steps

\int_{0}^{1} - 3 x (- 2 x^{2} - 1)^{3} d x

Take the constant out:

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

= - 3 \cdot \int_{0}^{1} x (- 2 x^{2} - 1)^{3} d x

Expand

x (- 2 x^{2} - 1)^{3} : - 8 x^{7} - 12 x^{5} - 6 x^{3} - x

= - 3 \cdot \int_{0}^{1} - 8 x^{7} - 12 x^{5} - 6 x^{3} - x 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

= - 3 (- \int_{0}^{1} 8 x^{7} d x - \int_{0}^{1} 12 x^{5} d x - \int_{0}^{1} 6 x^{3} d x - \int_{0}^{1} x d x)

\int_{0}^{1} 8 x^{7} d x = 1

\int_{0}^{1} 12 x^{5} d x = 2

\int_{0}^{1} 6 x^{3} d x = \frac{3}{2}

\int_{0}^{1} x d x = \frac{1}{2}

= - 3 (- 1 - 2 - \frac{3}{2} - \frac{1}{2})

Simplify

- 3 (- 1 - 2 - \frac{3}{2} - \frac{1}{2}) : 15

= 15

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What is the integral from 0 to 1 of-3x(-2x^2-1)^3 ?
The integral from 0 to 1 of-3x(-2x^2-1)^3 is 15