What is the integral from 0 to 7 of 2pix(7x-x^2) ?

The integral from 0 to 7 of 2pix(7x-x^2) is (2401pi)/6

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to 7 of 2pix(7x-x^2)

\int_{0}^{7} 2 π x (7 x - x^{2}) d x

\frac{2401 π}{6}

Decimal Notation

1257.16066 \dots

Solution steps

\int_{0}^{7} 2 π x (7 x - x^{2}) 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}^{7} x (7 x - x^{2}) d x

Expand

x (7 x - x^{2}) : 7 x^{2} - x^{3}

= 2 π \cdot \int_{0}^{7} 7 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 π (\int_{0}^{7} 7 x^{2} d x - \int_{0}^{7} x^{3} d x)

\int_{0}^{7} 7 x^{2} d x = \frac{2401}{3}

\int_{0}^{7} x^{3} d x = \frac{2401}{4}

= 2 π (\frac{2401}{3} - \frac{2401}{4})

Simplify

2 π (\frac{2401}{3} - \frac{2401}{4}) : \frac{2401 π}{6}

= \frac{2401 π}{6}

3 \frac{d y}{d x} + 5 y = 1

2 y^{^{'}} = e^{\frac{x}{2}} + y

\int 3^{2 - x} d x

\int \frac{1}{x ^{2} - 2 x - 63} d x

What is the integral from 0 to 7 of 2pix(7x-x^2) ?
The integral from 0 to 7 of 2pix(7x-x^2) is (2401pi)/6