What is the integral from 0 to 2 of-2x(2+x^5) ?

The integral from 0 to 2 of-2x(2+x^5) is -312/7

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integral from 0 to 2 of-2x(2+x^5)

\int_{0}^{2} - 2 x (2 + x^{5}) d x

- \frac{312}{7}

Decimal

- 44.57142 \dots

Solution steps

\int_{0}^{2} - 2 x (2 + x^{5}) d x

Take the constant out:

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

= - 2 \cdot \int_{0}^{2} x (2 + x^{5}) d x

Expand

x (2 + x^{5}) : 2 x + x^{6}

= - 2 \cdot \int_{0}^{2} 2 x + x^{6} 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

= - 2 (\int_{0}^{2} 2 x d x + \int_{0}^{2} x^{6} d x)

\int_{0}^{2} 2 x d x = 4

\int_{0}^{2} x^{6} d x = \frac{128}{7}

= - 2 (4 + \frac{128}{7})

Simplify

- 2 (4 + \frac{128}{7}) : - \frac{312}{7}

= - \frac{312}{7}

What is the integral from 0 to 2 of-2x(2+x^5) ?
The integral from 0 to 2 of-2x(2+x^5) is -312/7