What is the integral from-1 to 0 of 4x^2sqrt(4x^3+6) ?

The integral from-1 to 0 of 4x^2sqrt(4x^3+6) is (2(6sqrt(6)-2sqrt(2)))/9

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integral from-1 to 0 of 4x^2sqrt(4x^3+6)

\int_{- 1}^{0} 4 x^{2} 4 x^{3} + 6 d x

\frac{2 ( 6 6 - 2 2 )}{9}

Decimal

2.63744 \dots

Solution steps

\int_{- 1}^{0} 4 x^{2} 4 x^{3} + 6 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^{2} 4 x^{3} + 6 d x

Apply u-substitution

= 4 \cdot \int_{2}^{6} \frac{u ^{2}}{6} d u

Take the constant out:

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

= 4 \cdot \frac{1}{6} \cdot \int_{2}^{6} u^{2} d u

Apply the Power Rule

= 4 \cdot \frac{1}{6} [\frac{u ^{3}}{3}]_{2}^{6}

Simplify

4 \cdot \frac{1}{6} [\frac{u ^{3}}{3}]_{2}^{6} : \frac{2}{3} [\frac{u ^{3}}{3}]_{2}^{6}

= \frac{2}{3} [\frac{u ^{3}}{3}]_{2}^{6}

Compute the boundaries:

26 - \frac{2 2}{3}

= \frac{2}{3} (26 - \frac{2 2}{3})

Simplify

= \frac{2 ( 6 6 - 2 2 )}{9}

What is the integral from-1 to 0 of 4x^2sqrt(4x^3+6) ?
The integral from-1 to 0 of 4x^2sqrt(4x^3+6) is (2(6sqrt(6)-2sqrt(2)))/9