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

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

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

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

\frac{52}{3}

Decimal

17.33333 \dots

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= 3 \cdot \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

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

Apply the Power Rule

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

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

Simplify

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

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

Compute the boundaries:

\frac{52}{3}

= \frac{52}{3}

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