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

The integral from 3 to 5 of 3x(x+2) is 146

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

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

146

Solution steps

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

Take the constant out:

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

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

Expand

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

= 3 \cdot \int_{3}^{5} x^{2} + 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

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

\int_{3}^{5} x^{2} d x = \frac{98}{3}

\int_{3}^{5} 2 x d x = 16

= 3 (\frac{98}{3} + 16)

Simplify

3 (\frac{98}{3} + 16) : 146

= 146

What is the integral from 3 to 5 of 3x(x+2) ?
The integral from 3 to 5 of 3x(x+2) is 146