What is the integral of 2x^{2/3}(1+3/2 x^{5/6}) ?

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

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integral of 2x^{2/3}(1+3/2 x^{5/6})

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

2 (\frac{3}{5} x^{\frac{5}{3}} + \frac{3}{5} x^{\frac{5}{2}}) + C

Solution steps

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

Take the constant out:

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

= 2 \cdot \int x^{\frac{2}{3}} (1 + \frac{3}{2} x^{\frac{5}{6}}) d x

Expand

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

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

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

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

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

Add a constant to the solution

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

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