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

The integral from 0 to 1 of x^2(2(1-x)) is 1/6

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

\frac{1}{6}

Decimal

0.16666 \dots

Solution steps

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

Take the constant out:

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

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

Expand

x^{2} (1 - x) : x^{2} - x^{3}

= 2 \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

= 2 (\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}

= 2 (\frac{1}{3} - \frac{1}{4})

Simplify

2 (\frac{1}{3} - \frac{1}{4}) : \frac{1}{6}

= \frac{1}{6}

What is the integral from 0 to 1 of x^2(2(1-x)) ?
The integral from 0 to 1 of x^2(2(1-x)) is 1/6