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

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

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integral of ((x+1)/(x^2sqrt(x)))

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

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

Solution steps

\int \frac{x + 1}{x ^{2} x} d x

Simplify

x^{2} x : x^{\frac{5}{2}}

= \int \frac{x + 1}{x ^{\frac{5}{2}}} d x

Expand

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

= \int \frac{1}{x ^{\frac{3}{2}}} + \frac{1}{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

= \int \frac{1}{x ^{\frac{3}{2}}} d x + \int \frac{1}{x ^{\frac{5}{2}}} d x

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

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

= - \frac{2}{x} - \frac{2}{3 x ^{\frac{3}{2}}}

Add a constant to the solution

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

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