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

The integral from-1 to 2 of 2x(x^2+2)^2 is 63

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

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

63

Solution steps

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

Take the constant out:

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

= 2 \cdot \int_{- 1}^{2} x (x^{2} + 2)^{2} d x

Expand

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

= 2 \cdot \int_{- 1}^{2} 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

= 2 (\int_{- 1}^{2} x^{5} d x + \int_{- 1}^{2} 4 x^{3} d x + \int_{- 1}^{2} 4 x d x)

\int_{- 1}^{2} x^{5} d x = \frac{21}{2}

\int_{- 1}^{2} 4 x^{3} d x = 15

\int_{- 1}^{2} 4 x d x = 6

= 2 (\frac{21}{2} + 15 + 6)

Simplify

2 (\frac{21}{2} + 15 + 6) : 63

= 63

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What is the integral from-1 to 2 of 2x(x^2+2)^2 ?
The integral from-1 to 2 of 2x(x^2+2)^2 is 63