What is the integral from 0 to 2 of 2pix(3+(x^2)/3) ?

The integral from 0 to 2 of 2pix(3+(x^2)/3) is (44pi)/3

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integral from 0 to 2 of 2pix(3+(x^2)/3)

\int_{0}^{2} 2 π x (3 + \frac{x ^{2}}{3}) d x

\frac{44 π}{3}

Decimal

46.07669 \dots

Solution steps

\int_{0}^{2} 2 π x (3 + \frac{x ^{2}}{3}) 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 (3 + \frac{x ^{2}}{3}) d x

Apply Integration By Parts

= 2 π [\frac{1}{6} x^{2} (9 + x^{2}) - \int \frac{x ^{3}}{3} d x]_{0}^{2}

\int \frac{x ^{3}}{3} d x = \frac{x ^{4}}{12}

= 2 π [\frac{1}{6} x^{2} (9 + x^{2}) - \frac{x ^{4}}{12}]_{0}^{2}

Simplify

2 π [\frac{1}{6} x^{2} (9 + x^{2}) - \frac{x ^{4}}{12}]_{0}^{2} : 2 π [\frac{1}{12} (18 x^{2} + x^{4})]_{0}^{2}

= 2 π [\frac{1}{12} (18 x^{2} + x^{4})]_{0}^{2}

Compute the boundaries:

\frac{22}{3}

= 2 π \frac{22}{3}

Simplify

= \frac{44 π}{3}

What is the integral from 0 to 2 of 2pix(3+(x^2)/3) ?
The integral from 0 to 2 of 2pix(3+(x^2)/3) is (44pi)/3