What is the integral from 0 to x of 20(3/2 x)^{1/3} ?

The integral from 0 to x of 20(3/2 x)^{1/3} is (15*12^{1/3}x^{4/3})/2

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integral from 0 to x of 20(3/2 x)^{1/3}

\int_{0}^{x} 20 (\frac{3}{2} x)^{\frac{1}{3}} d x

\frac{15 \cdot 1 2 ^{\frac{1}{3}} x ^{\frac{4}{3}}}{2}

Solution steps

\int_{0}^{x} 20 (\frac{3}{2} x)^{\frac{1}{3}} d x

Take the constant out:

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

= 20 \cdot \int_{0}^{x} (\frac{3}{2} x)^{\frac{1}{3}} d x

Simplify

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

= 20 \cdot \int_{0}^{x} (\frac{3}{2})^{\frac{1}{3}} x^{\frac{1}{3}} d x

Take the constant out:

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

= 20 (\frac{3}{2})^{\frac{1}{3}} \cdot \int_{0}^{x} x^{\frac{1}{3}} d x

Apply the Power Rule

: [\frac{3}{4} x^{\frac{4}{3}}]_{0}^{x}

= 20 (\frac{3}{2})^{\frac{1}{3}} [\frac{3}{4} x^{\frac{4}{3}}]_{0}^{x}

Simplify

20 (\frac{3}{2})^{\frac{1}{3}} [\frac{3}{4} x^{\frac{4}{3}}]_{0}^{x} : 10 \cdot 1 2^{\frac{1}{3}} [\frac{3}{4} x^{\frac{4}{3}}]_{0}^{x}

= 10 \cdot 1 2^{\frac{1}{3}} [\frac{3}{4} x^{\frac{4}{3}}]_{0}^{x}

Compute the boundaries:

\frac{3}{4} x^{\frac{4}{3}}

= 10 \cdot 1 2^{\frac{1}{3}} \cdot \frac{3}{4} x^{\frac{4}{3}}

Simplify

= \frac{15 \cdot 1 2 ^{\frac{1}{3}} x ^{\frac{4}{3}}}{2}

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What is the integral from 0 to x of 20(3/2 x)^{1/3} ?
The integral from 0 to x of 20(3/2 x)^{1/3} is (15*12^{1/3}x^{4/3})/2