What is the integral from 0 to 1 of 4x^3sqrt(1+9x^4) ?

The integral from 0 to 1 of 4x^3sqrt(1+9x^4) is (20sqrt(10)-2)/(27)

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integral from 0 to 1 of 4x^3sqrt(1+9x^4)

\int_{0}^{1} 4 x^{3} 1 + 9 x^{4} d x

\frac{20 10 - 2}{27}

Solution steps

\int_{0}^{1} 4 x^{3} 1 + 9 x^{4} d x

Take the constant out:

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

= 4 \cdot \int_{0}^{1} x^{3} 1 + 9 x^{4} d x

Apply u-substitution

: \int_{1}^{10} \frac{u}{36} d u

= 4 \cdot \int_{1}^{10} \frac{u}{36} d u

Take the constant out:

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

= 4 \cdot \frac{1}{36} \cdot \int_{1}^{10} u d u

Apply the Power Rule

: [\frac{2}{3} u^{\frac{3}{2}}]_{1}^{10}

= 4 \cdot \frac{1}{36} [\frac{2}{3} u^{\frac{3}{2}}]_{1}^{10}

Simplify

4 \cdot \frac{1}{36} [\frac{2}{3} u^{\frac{3}{2}}]_{1}^{10} : \frac{1}{9} [\frac{2}{3} u^{\frac{3}{2}}]_{1}^{10}

= \frac{1}{9} [\frac{2}{3} u^{\frac{3}{2}}]_{1}^{10}

Compute the boundaries:

\frac{20 10}{3} - \frac{2}{3}

= \frac{1}{9} (\frac{20 10}{3} - \frac{2}{3})

Simplify

= \frac{20 10 - 2}{27}

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What is the integral from 0 to 1 of 4x^3sqrt(1+9x^4) ?
The integral from 0 to 1 of 4x^3sqrt(1+9x^4) is (20sqrt(10)-2)/(27)