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

The integral from 1 to 3 of 3x(x^2+5) is 120

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

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

120

Solution steps

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

Take the constant out:

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

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

Expand

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

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

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

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

\int_{1}^{3} 5 x d x = 20

= 3 (20 + 20)

Simplify

= 120

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What is the integral from 1 to 3 of 3x(x^2+5) ?
The integral from 1 to 3 of 3x(x^2+5) is 120