What is the integral from 0 to 1 of 12x(x-x^2) ?

The integral from 0 to 1 of 12x(x-x^2) is 1

Popular >

integral from 0 to 1 of 12x(x-x^2)

\int_{0}^{1} 12 x (x - x^{2}) d x

1

Solution steps

\int_{0}^{1} 12 x (x - x^{2}) d x

Take the constant out:

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

= 12 \cdot \int_{0}^{1} x (x - x^{2}) d x

Expand

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

= 12 \cdot \int_{0}^{1} 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

= 12 (\int_{0}^{1} x^{2} d x - \int_{0}^{1} x^{3} d x)

\int_{0}^{1} x^{2} d x = \frac{1}{3}

\int_{0}^{1} x^{3} d x = \frac{1}{4}

= 12 (\frac{1}{3} - \frac{1}{4})

Simplify

12 (\frac{1}{3} - \frac{1}{4}) : 1

= 1

\int_{2}^{4} \frac{x - 2}{x} d x

\int_{- 1}^{0} (2 t + 2)^{2} d t

\int_{1}^{\infty} x e^{- a x} d x

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

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

What is the integral from 0 to 1 of 12x(x-x^2) ?
The integral from 0 to 1 of 12x(x-x^2) is 1