What is the integral from 0 to 8 of 2pix(x^{3/2}) ?

The integral from 0 to 8 of 2pix(x^{3/2}) is (4096*2^{1/2}pi)/7

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integral from 0 to 8 of 2pix(x^{3/2})

\int_{0}^{8} 2 π x (x^{\frac{3}{2}}) d x

\frac{4096 \cdot 2 ^{\frac{1}{2}} π}{7}

Decimal Notation

2599.72121 \dots

Solution steps

\int_{0}^{8} 2 π x x^{\frac{3}{2}} 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}^{8} x x^{\frac{3}{2}} d x

Simplify

x x^{\frac{3}{2}} : x^{\frac{5}{2}}

= 2 π \cdot \int_{0}^{8} x^{\frac{5}{2}} d x

Apply the Power Rule

: [\frac{2}{7} x^{\frac{7}{2}}]_{0}^{8}

= 2 π [\frac{2}{7} x^{\frac{7}{2}}]_{0}^{8}

Compute the boundaries:

\frac{2048 \cdot 2 ^{\frac{1}{2}}}{7}

= 2 π \frac{2048 \cdot 2 ^{\frac{1}{2}}}{7}

Simplify

= \frac{4096 \cdot 2 ^{\frac{1}{2}} π}{7}

\int 8 x sec^{2} (x) d x

\int x sin (2 a x) d x

y = (1 + 6 x^{2}) (x - x^{2})

n = 1 \sum \infty \frac{5}{n ^{4}}

x \to \infty lim (\frac{x}{3 x})

What is the integral from 0 to 8 of 2pix(x^{3/2}) ?
The integral from 0 to 8 of 2pix(x^{3/2}) is (4096*2^{1/2}pi)/7