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

The integral from 0 to 2 of (x+2x^2-x^3) is 10/3

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

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

\frac{10}{3}

Decimal

3.33333 \dots

Solution steps

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

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

\int_{0}^{2} 2 x^{2} d x = \frac{16}{3}

\int_{0}^{2} x^{3} d x = 4

= 2 + \frac{16}{3} - 4

Simplify

2 + \frac{16}{3} - 4 : \frac{10}{3}

= \frac{10}{3}

What is the integral from 0 to 2 of (x+2x^2-x^3) ?
The integral from 0 to 2 of (x+2x^2-x^3) is 10/3