What is the integral from 0 to 1 of 2pix(x^3-x^4) ?

The integral from 0 to 1 of 2pix(x^3-x^4) is pi/(15)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to 1 of 2pix(x^3-x^4)

\int_{0}^{1} 2 π x (x^{3} - x^{4}) d x

\frac{π}{15}

Decimal Notation

0.20943 \dots

Solution steps

\int_{0}^{1} 2 π x (x^{3} - x^{4}) d x

Take the constant out:

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

= 2 π \cdot \int_{0}^{1} x (x^{3} - x^{4}) d x

Expand

x (x^{3} - x^{4}) : x^{4} - x^{5}

= 2 π \cdot \int_{0}^{1} x^{4} - x^{5} 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 π (\int_{0}^{1} x^{4} d x - \int_{0}^{1} x^{5} d x)

\int_{0}^{1} x^{4} d x = \frac{1}{5}

\int_{0}^{1} x^{5} d x = \frac{1}{6}

= 2 π (\frac{1}{5} - \frac{1}{6})

Simplify

2 π (\frac{1}{5} - \frac{1}{6}) : \frac{π}{15}

= \frac{π}{15}

\int \frac{x}{( x - 3 ) ^{2}} d x

\frac{\partial}{\partial x} (\frac{y ^{2}}{4 - x ^{2}})

\int \frac{1}{- x ^{2} + 6 x - 10} d x

\frac{d}{d x} ((sinh (x))^{2 x^{2} + 2 x})

\int \frac{( ln ( x ) ) ^{2}}{2} d x

What is the integral from 0 to 1 of 2pix(x^3-x^4) ?
The integral from 0 to 1 of 2pix(x^3-x^4) is pi/(15)