What is the integral from 1 to 2 of 4x(x^2+2) ?

The integral from 1 to 2 of 4x(x^2+2) is 27

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integral from 1 to 2 of 4x(x^2+2)

\int_{1}^{2} 4 x (x^{2} + 2) d x

27

Solution steps

\int_{1}^{2} 4 x (x^{2} + 2) d x

Take the constant out:

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

= 4 \cdot \int_{1}^{2} x (x^{2} + 2) d x

Expand

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

= 4 \cdot \int_{1}^{2} x^{3} + 2 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

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

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

\int_{1}^{2} 2 x d x = 3

= 4 (\frac{15}{4} + 3)

Simplify

4 (\frac{15}{4} + 3) : 27

= 27

What is the integral from 1 to 2 of 4x(x^2+2) ?
The integral from 1 to 2 of 4x(x^2+2) is 27