What is the integral from 0 to 3 of x(3x-x^2) ?

The integral from 0 to 3 of x(3x-x^2) is 27/4

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integral from 0 to 3 of x(3x-x^2)

\int_{0}^{3} x (3 x - x^{2}) d x

\frac{27}{4}

Decimal Notation

6.75

Solution steps

\int_{0}^{3} x (3 x - x^{2}) d x

Expand

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

= \int_{0}^{3} 3 x^{2} - x^{3} 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

= \int_{0}^{3} 3 x^{2} d x - \int_{0}^{3} x^{3} d x

\int_{0}^{3} 3 x^{2} d x = 27

\int_{0}^{3} x^{3} d x = \frac{81}{4}

= 27 - \frac{81}{4}

Simplify

27 - \frac{81}{4} : \frac{27}{4}

= \frac{27}{4}

\int_{- 1}^{1} ∣ 5^{x} - 2^{x} ∣ d x

\int_{0}^{a} e^{- a x} d x

\int_{1}^{2} \frac{x}{x + y} d y

\int_{0}^{1} (1 + 2 x)^{3} d x

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

What is the integral from 0 to 3 of x(3x-x^2) ?
The integral from 0 to 3 of x(3x-x^2) is 27/4