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

The integral from 0 to 1 of x(x^2+2)^2 is 19/6

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

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

\frac{19}{6}

Decimal Notation

3.16666 \dots

Solution steps

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

Expand

x (x^{2} + 2)^{2} : x^{5} + 4 x^{3} + 4 x

= \int_{0}^{1} x^{5} + 4 x^{3} + 4 x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{1} x^{5} d x + \int_{0}^{1} 4 x^{3} d x + \int_{0}^{1} 4 x d x

\int_{0}^{1} x^{5} d x = \frac{1}{6}

\int_{0}^{1} 4 x^{3} d x = 1

\int_{0}^{1} 4 x d x = 2

= \frac{1}{6} + 1 + 2

Simplify

\frac{1}{6} + 1 + 2 : \frac{19}{6}

= \frac{19}{6}

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What is the integral from 0 to 1 of x(x^2+2)^2 ?
The integral from 0 to 1 of x(x^2+2)^2 is 19/6