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

The integral from 0 to 4 of 2pix(2sqrt(x)) is (256pi)/5

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

\int_{0}^{4} 2 π x (2 x) d x

\frac{256 π}{5}

Decimal

160.84954 \dots

Solution steps

\int_{0}^{4} 2 π x \cdot 2 x d x

Take the constant out:

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

= 2 π 2 \cdot \int_{0}^{4} x x d x

Simplify

= 4 π \cdot \int_{0}^{4} x x d x

Simplify

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

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

Apply the Power Rule

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

Compute the boundaries:

\frac{64}{5}

= 4 π \frac{64}{5}

Simplify

= \frac{256 π}{5}

What is the integral from 0 to 4 of 2pix(2sqrt(x)) ?
The integral from 0 to 4 of 2pix(2sqrt(x)) is (256pi)/5