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

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

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

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

\frac{4 2 - 2}{9}

Decimal

0.40631 \dots

Solution steps

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

Apply u-substitution

: \int_{1}^{2} \frac{u}{3} d u

= \int_{1}^{2} \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}^{2} u d u

Apply the Power Rule

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

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

Compute the boundaries:

\frac{4 2}{3} - \frac{2}{3}

= \frac{1}{3} (\frac{4 2}{3} - \frac{2}{3})

Simplify

= \frac{4 2 - 2}{9}

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