What is the integral of sqrt(x)(x^4+1/(x^2)) ?

The integral of sqrt(x)(x^4+1/(x^2)) is -2/(sqrt(x))+2/11 x^{11/2}+C

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integral of sqrt(x)(x^4+1/(x^2))

\int x (x^{4} + \frac{1}{x ^{2}}) d x

- \frac{2}{x} + \frac{2}{11} x^{\frac{11}{2}} + C

Solution steps

\int x (x^{4} + \frac{1}{x ^{2}}) d x

Expand

x (x^{4} + \frac{1}{x ^{2}}) : \frac{1}{x ^{\frac{3}{2}}} + x^{\frac{9}{2}}

= \int \frac{1}{x ^{\frac{3}{2}}} + x^{\frac{9}{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 x^{\frac{9}{2}} d x

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

\int x^{\frac{9}{2}} d x = \frac{2}{11} x^{\frac{11}{2}}

= - \frac{2}{x} + \frac{2}{11} x^{\frac{11}{2}}

Add a constant to the solution

= - \frac{2}{x} + \frac{2}{11} x^{\frac{11}{2}} + C

What is the integral of sqrt(x)(x^4+1/(x^2)) ?
The integral of sqrt(x)(x^4+1/(x^2)) is -2/(sqrt(x))+2/11 x^{11/2}+C