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

The integral of 6/5 x^{5/3}(1+x^{5/6}) is 6/5 (3/8 x^{8/3}+2/7 x^{7/2})+C

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

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

\frac{6}{5} (\frac{3}{8} x^{\frac{8}{3}} + \frac{2}{7} x^{\frac{7}{2}}) + C

Solution steps

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

Take the constant out:

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

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

Expand

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

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

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

\int x^{\frac{5}{3}} d x = \frac{3}{8} x^{\frac{8}{3}}

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

= \frac{6}{5} (\frac{3}{8} x^{\frac{8}{3}} + \frac{2}{7} x^{\frac{7}{2}})

Add a constant to the solution

= \frac{6}{5} (\frac{3}{8} x^{\frac{8}{3}} + \frac{2}{7} x^{\frac{7}{2}}) + C

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What is the integral of 6/5 x^{5/3}(1+x^{5/6}) ?
The integral of 6/5 x^{5/3}(1+x^{5/6}) is 6/5 (3/8 x^{8/3}+2/7 x^{7/2})+C