What is the integral from 0 to 4 of 2pix(sqrt(x)) ?

The integral from 0 to 4 of 2pix(sqrt(x)) is (128pi)/5

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integral from 0 to 4 of 2pix(sqrt(x))

\int_{0}^{4} 2 π x (x) d x

\frac{128 π}{5}

Decimal

80.42477 \dots

Solution steps

\int_{0}^{4} 2 π x 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_{0}^{4} x x d x

Simplify

x x : x^{\frac{3}{2}}

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

Apply the Power Rule

: [\frac{2}{5} x^{\frac{5}{2}}]_{0}^{4}

= 2 π [\frac{2}{5} x^{\frac{5}{2}}]_{0}^{4}

Compute the boundaries:

\frac{64}{5}

= 2 π \frac{64}{5}

Simplify

= \frac{128 π}{5}

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n = 1 \sum \infty \frac{1}{3 n + 1}

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What is the integral from 0 to 4 of 2pix(sqrt(x)) ?
The integral from 0 to 4 of 2pix(sqrt(x)) is (128pi)/5