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

The integral from 0 to 2 of x^3sqrt(x+2) is (1664+512sqrt(2))/(315)

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

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

\frac{1664 + 512 2}{315}

Decimal

7.58119 \dots

Solution steps

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

Apply Integration By Parts

: [\frac{2}{3} x^{3} (x + 2)^{\frac{3}{2}} - \int 2 x^{2} (x + 2)^{\frac{3}{2}} d x]_{0}^{2}

= [\frac{2}{3} x^{3} (x + 2)^{\frac{3}{2}} - \int 2 x^{2} (x + 2)^{\frac{3}{2}} d x]_{0}^{2}

\int 2 x^{2} (x + 2)^{\frac{3}{2}} d x = 2 (\frac{2}{9} (x + 2)^{\frac{9}{2}} - \frac{8}{7} (x + 2)^{\frac{7}{2}} + \frac{8}{5} (x + 2)^{\frac{5}{2}})

= [\frac{2}{3} x^{3} (x + 2)^{\frac{3}{2}} - 2 (\frac{2}{9} (x + 2)^{\frac{9}{2}} - \frac{8}{7} (x + 2)^{\frac{7}{2}} + \frac{8}{5} (x + 2)^{\frac{5}{2}})]_{0}^{2}

Compute the boundaries:

\frac{1664}{315} + \frac{512 2}{315}

= \frac{1664}{315} + \frac{512 2}{315}

Simplify

= \frac{1664 + 512 2}{315}

\int \frac{5 x ^{2} - 6}{x ^{2} - 2 x - 48} d x

\frac{d}{d t} (12 t^{2} sin (t))

y = 3 a - b^{2} x^{4}

x \to - 2 lim ((x + 2)^{5} (2 x^{2}))

\frac{\partial}{\partial x} (\frac{- x}{y ^{2}})

What is the integral from 0 to 2 of x^3sqrt(x+2) ?
The integral from 0 to 2 of x^3sqrt(x+2) is (1664+512sqrt(2))/(315)