What is the integral from 0 to 8 of x\sqrt[3]{x} ?

The integral from 0 to 8 of x\sqrt[3]{x} is 384/7

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integral from 0 to 8 of x\sqrt[3]{x}

\int_{0}^{8} x 3 x d x

\frac{384}{7}

Decimal

54.85714 \dots

Solution steps

\int_{0}^{8} x 3 x d x

Apply Integration By Parts

: [\frac{1}{2} x^{2} 3 x - \int \frac{1}{6} x^{\frac{4}{3}} d x]_{0}^{8}

= [\frac{1}{2} x^{2} 3 x - \int \frac{1}{6} x^{\frac{4}{3}} d x]_{0}^{8}

\int \frac{1}{6} x^{\frac{4}{3}} d x = \frac{1}{14} x^{\frac{7}{3}}

= [\frac{1}{2} x^{2} 3 x - \frac{1}{14} x^{\frac{7}{3}}]_{0}^{8}

Simplify

[\frac{1}{2} x^{2} 3 x - \frac{1}{14} x^{\frac{7}{3}}]_{0}^{8} : [\frac{3}{7} x^{2} 3 x]_{0}^{8}

= [\frac{3}{7} x^{2} 3 x]_{0}^{8}

Compute the boundaries:

\frac{384}{7}

= \frac{384}{7}

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What is the integral from 0 to 8 of x\sqrt[3]{x} ?
The integral from 0 to 8 of x\sqrt[3]{x} is 384/7