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

The integral of sqrt(x)(x^{1/3}-x^{2/5}) is 6/11 x^{11/6}-10/19 x^{19/10}+C

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

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

\frac{6}{11} x^{\frac{11}{6}} - \frac{10}{19} x^{\frac{19}{10}} + C

Solution steps

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

Expand

x (x^{\frac{1}{3}} - x^{\frac{2}{5}}) : x^{\frac{5}{6}} - x^{\frac{9}{10}}

= \int x^{\frac{5}{6}} - x^{\frac{9}{10}} 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

= \int x^{\frac{5}{6}} d x - \int x^{\frac{9}{10}} d x

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

\int x^{\frac{9}{10}} d x = \frac{10}{19} x^{\frac{19}{10}}

= \frac{6}{11} x^{\frac{11}{6}} - \frac{10}{19} x^{\frac{19}{10}}

Add a constant to the solution

= \frac{6}{11} x^{\frac{11}{6}} - \frac{10}{19} x^{\frac{19}{10}} + C

\int_{0}^{\infty} sin (x) d x

\frac{4 x + 1}{x}

\frac{d y}{d x} = \frac{x}{1 + x ^{2}}

\frac{1}{3} y (y - 3)

What is the integral of sqrt(x)(x^{1/3}-x^{2/5}) ?
The integral of sqrt(x)(x^{1/3}-x^{2/5}) is 6/11 x^{11/6}-10/19 x^{19/10}+C