What is the integral from 1 to 3 of 5x(x^2+4) ?

The integral from 1 to 3 of 5x(x^2+4) is 180

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integral from 1 to 3 of 5x(x^2+4)

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

180

Solution steps

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

Take the constant out:

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

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

Expand

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

= 5 \cdot \int_{1}^{3} 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

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

\int_{1}^{3} x^{3} d x = 20

\int_{1}^{3} 4 x d x = 16

= 5 (20 + 16)

Simplify

= 180

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What is the integral from 1 to 3 of 5x(x^2+4) ?
The integral from 1 to 3 of 5x(x^2+4) is 180