What is the integral of (3/2 x+5)^{1/3} ?

The integral of (3/2 x+5)^{1/3} is 1/(4*2^{1/3)}(3x+10)^{4/3}+C

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integral of (3/2 x+5)^{1/3}

\int (\frac{3}{2} x + 5)^{\frac{1}{3}} d x

\frac{1}{4 \cdot 2 ^{\frac{1}{3}}} (3 x + 10)^{\frac{4}{3}} + C

Solution steps

\int (\frac{3}{2} x + 5)^{\frac{1}{3}} d x

Apply u-substitution

: \int \frac{2 u ^{\frac{1}{3}}}{3} d u

= \int \frac{2 u ^{\frac{1}{3}}}{3} d u

Take the constant out:

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

= \frac{2}{3} \cdot \int u^{\frac{1}{3}} d u

Apply the Power Rule

: \frac{3}{4} u^{\frac{4}{3}}

= \frac{2}{3} \cdot \frac{3}{4} u^{\frac{4}{3}}

Substitute back

u = \frac{3}{2} x + 5

= \frac{2}{3} \cdot \frac{3}{4} (\frac{3}{2} x + 5)^{\frac{4}{3}}

Simplify

\frac{2}{3} \cdot \frac{3}{4} (\frac{3}{2} x + 5)^{\frac{4}{3}} : \frac{1}{4 \cdot 2 ^{\frac{1}{3}}} (3 x + 10)^{\frac{4}{3}}

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

Add a constant to the solution

= \frac{1}{4 \cdot 2 ^{\frac{1}{3}}} (3 x + 10)^{\frac{4}{3}} + C

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What is the integral of (3/2 x+5)^{1/3} ?
The integral of (3/2 x+5)^{1/3} is 1/(4*2^{1/3)}(3x+10)^{4/3}+C