What is the integral from-3 to-1 of 2x(5-x^2)^3 ?

The integral from-3 to-1 of 2x(5-x^2)^3 is 0

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integral from-3 to-1 of 2x(5-x^2)^3

\int_{- 3}^{- 1} 2 x (5 - x^{2})^{3} d x

0

Solution steps

\int_{- 3}^{- 1} 2 x (5 - x^{2})^{3} d x

Take the constant out:

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

= 2 \cdot \int_{- 3}^{- 1} x (5 - x^{2})^{3} d x

Apply u-substitution

= 2 \cdot \int_{- 4}^{4} - \frac{u ^{3}}{2} d u

Take the constant out:

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

= 2 (- \frac{1}{2} \cdot \int_{- 4}^{4} u^{3} d u)

Apply the Power Rule

= 2 (- \frac{1}{2} [\frac{u ^{4}}{4}]_{- 4}^{4})

Simplify

2 (- \frac{1}{2} [\frac{u ^{4}}{4}]_{- 4}^{4}) : - [\frac{u ^{4}}{4}]_{- 4}^{4}

= - [\frac{u ^{4}}{4}]_{- 4}^{4}

Compute the boundaries:

0

= - 0

Simplify

= 0

What is the integral from-3 to-1 of 2x(5-x^2)^3 ?
The integral from-3 to-1 of 2x(5-x^2)^3 is 0