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

The integral from 0 to 2 of 2pix(1+(x^2)/4) is 6pi

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integral from 0 to 2 of 2pix(1+(x^2)/4)

\int_{0}^{2} 2 π x (1 + \frac{x ^{2}}{4}) d x

6 π

Decimal

18.84955 \dots

Solution steps

\int_{0}^{2} 2 π x (1 + \frac{x ^{2}}{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}^{2} x (1 + \frac{x ^{2}}{4}) d x

Expand

x (1 + \frac{x ^{2}}{4}) : x + \frac{x ^{3}}{4}

= 2 π \cdot \int_{0}^{2} x + \frac{x ^{3}}{4} 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}^{2} x d x + \int_{0}^{2} \frac{x ^{3}}{4} d x)

\int_{0}^{2} x d x = 2

\int_{0}^{2} \frac{x ^{3}}{4} d x = 1

= 2 π (2 + 1)

Simplify

= 6 π

What is the integral from 0 to 2 of 2pix(1+(x^2)/4) ?
The integral from 0 to 2 of 2pix(1+(x^2)/4) is 6pi