What is the integral of x(sqrt(x)+2/(x^3)) ?

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

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integral of x(sqrt(x)+2/(x^3))

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

- \frac{2}{x} + \frac{2}{5} x^{\frac{5}{2}} + C

Solution steps

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

Expand

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

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

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

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

= - \frac{2}{x} + \frac{2}{5} x^{\frac{5}{2}}

Add a constant to the solution

= - \frac{2}{x} + \frac{2}{5} x^{\frac{5}{2}} + C

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