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

The integral of 2sqrt(x)(x+2x^{1/3}) is 2(2/5 x^{5/2}+12/11 x^{11/6})+C

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

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

2 (\frac{2}{5} x^{\frac{5}{2}} + \frac{12}{11} x^{\frac{11}{6}}) + C

Solution steps

\int 2 x (x + 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

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

Expand

x (x + 2 x^{\frac{1}{3}}) : x^{\frac{3}{2}} + 2 x^{\frac{5}{6}}

= 2 \cdot \int x^{\frac{3}{2}} + 2 x^{\frac{5}{6}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int x^{\frac{3}{2}} d x + \int 2 x^{\frac{5}{6}} d x)

\int x^{\frac{3}{2}} d x = \frac{2}{5} x^{\frac{5}{2}}

\int 2 x^{\frac{5}{6}} d x = \frac{12}{11} x^{\frac{11}{6}}

= 2 (\frac{2}{5} x^{\frac{5}{2}} + \frac{12}{11} x^{\frac{11}{6}})

Add a constant to the solution

= 2 (\frac{2}{5} x^{\frac{5}{2}} + \frac{12}{11} x^{\frac{11}{6}}) + C

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What is the integral of 2sqrt(x)(x+2x^{1/3}) ?
The integral of 2sqrt(x)(x+2x^{1/3}) is 2(2/5 x^{5/2}+12/11 x^{11/6})+C