What is the integral of (x^{(-3)/2}+x-7)/(sqrt(x)) ?

The integral of (x^{(-3)/2}+x-7)/(sqrt(x)) is -1/x+2/3 x^{3/2}-14sqrt(x)+C

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integral of (x^{(-3)/2}+x-7)/(sqrt(x))

\int \frac{x ^{\frac{- 3}{2}} + x - 7}{x} d x

- \frac{1}{x} + \frac{2}{3} x^{\frac{3}{2}} - 14 x + C

Solution steps

\int \frac{x ^{\frac{- 3}{2}} + x - 7}{x} d x

\frac{x ^{\frac{- 3}{2}} + x - 7}{x} = (\frac{x ^{\frac{- 3}{2}}}{x} + \frac{x}{x} - \frac{7}{x})

= \int \frac{x ^{\frac{- 3}{2}}}{x} + \frac{x}{x} - \frac{7}{x} 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 \frac{x ^{\frac{- 3}{2}}}{x} d x + \int \frac{x}{x} d x - \int \frac{7}{x} d x

\int \frac{x ^{\frac{- 3}{2}}}{x} d x = - \frac{1}{x}

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

\int \frac{7}{x} d x = 14 x

= - \frac{1}{x} + \frac{2}{3} x^{\frac{3}{2}} - 14 x

Add a constant to the solution

= - \frac{1}{x} + \frac{2}{3} x^{\frac{3}{2}} - 14 x + C

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What is the integral of (x^{(-3)/2}+x-7)/(sqrt(x)) ?
The integral of (x^{(-3)/2}+x-7)/(sqrt(x)) is -1/x+2/3 x^{3/2}-14sqrt(x)+C