What is the integral from-1 to 0 of 4x(-4x^2+5)^3 ?

The integral from-1 to 0 of 4x(-4x^2+5)^3 is -78

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integral from-1 to 0 of 4x(-4x^2+5)^3

\int_{- 1}^{0} 4 x (- 4 x^{2} + 5)^{3} d x

- 78

Solution steps

\int_{- 1}^{0} 4 x (- 4 x^{2} + 5)^{3} d x

Take the constant out:

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

= 4 \cdot \int_{- 1}^{0} x (- 4 x^{2} + 5)^{3} d x

Apply u-substitution

: \int_{1}^{5} - \frac{u ^{3}}{8} d u

= 4 \cdot \int_{1}^{5} - \frac{u ^{3}}{8} d u

Take the constant out:

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

= 4 (- \frac{1}{8} \cdot \int_{1}^{5} u^{3} d u)

Apply the Power Rule

: [\frac{u ^{4}}{4}]_{1}^{5}

= 4 (- \frac{1}{8} [\frac{u ^{4}}{4}]_{1}^{5})

Simplify

4 (- \frac{1}{8} [\frac{u ^{4}}{4}]_{1}^{5}) : - \frac{1}{2} [\frac{u ^{4}}{4}]_{1}^{5}

= - \frac{1}{2} [\frac{u ^{4}}{4}]_{1}^{5}

Compute the boundaries:

156

= - \frac{1}{2} \cdot 156

Simplify

= - 78

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What is the integral from-1 to 0 of 4x(-4x^2+5)^3 ?
The integral from-1 to 0 of 4x(-4x^2+5)^3 is -78