What is the integral from 1 to 4 of 2pix(1+sqrt(x)) ?

The integral from 1 to 4 of 2pix(1+sqrt(x)) is (199pi)/5

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 1 to 4 of 2pix(1+sqrt(x))

\int_{1}^{4} 2 π x (1 + x) d x

\frac{199 π}{5}

Decimal

125.03538 \dots

Solution steps

\int_{1}^{4} 2 π x (1 + x) d x

Take the constant out:

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

= 2 π \cdot \int_{1}^{4} x (1 + x) d x

Expand

x (1 + x) : x + x^{\frac{3}{2}}

= 2 π \cdot \int_{1}^{4} x + x^{\frac{3}{2}} 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_{1}^{4} x d x + \int_{1}^{4} x^{\frac{3}{2}} d x)

\int_{1}^{4} x d x = \frac{15}{2}

\int_{1}^{4} x^{\frac{3}{2}} d x = \frac{62}{5}

= 2 π (\frac{15}{2} + \frac{62}{5})

Simplify

2 π (\frac{15}{2} + \frac{62}{5}) : \frac{199 π}{5}

= \frac{199 π}{5}

\int_{- 1}^{0} \frac{1}{x} d x

\int_{1}^{4} 10 x d x

\int_{0}^{2} 2 sin (\frac{π x}{2}) d x

\int_{0}^{10} \frac{1}{2} x d x

\int_{0}^{9} 4 x d x

What is the integral from 1 to 4 of 2pix(1+sqrt(x)) ?
The integral from 1 to 4 of 2pix(1+sqrt(x)) is (199pi)/5