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

The integral from 0 to 2 of (x^2sqrt(1+x^3)) is 52/9

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

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

\frac{52}{9}

Solution steps

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

Apply u-substitution

: \int_{1}^{9} \frac{u}{3} d u

= \int_{1}^{9} \frac{u}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int_{1}^{9} u d u

Apply the Power Rule

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

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

Compute the boundaries:

\frac{52}{3}

= \frac{1}{3} \cdot \frac{52}{3}

Simplify

= \frac{52}{9}

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What is the integral from 0 to 2 of (x^2sqrt(1+x^3)) ?
The integral from 0 to 2 of (x^2sqrt(1+x^3)) is 52/9