What is the integral of sqrt(x)(x^3-1/x) ?

The integral of sqrt(x)(x^3-1/x) is 2/9 x^4sqrt(x)-2sqrt(x)+C

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integral of sqrt(x)(x^3-1/x)

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

\frac{2}{9} x^{4} x - 2 x + C

Solution steps

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

Apply Integration By Parts

= x^{4} x - x - \int \frac{x ^{4} + 6 x x ^{\frac{7}{2}} + 1}{2 x ^{\frac{1}{2}}} d x

\int \frac{x ^{4} + 6 x x ^{\frac{7}{2}} + 1}{2 x ^{\frac{1}{2}}} d x = \frac{1}{9} x (7 x^{4} + 9)

= x^{4} x - x - \frac{1}{9} x (7 x^{4} + 9)

Simplify

x^{4} x - x - \frac{1}{9} x (7 x^{4} + 9) : \frac{2}{9} x^{4} x - 2 x

= \frac{2}{9} x^{4} x - 2 x

Add a constant to the solution

= \frac{2}{9} x^{4} x - 2 x + C

What is the integral of sqrt(x)(x^3-1/x) ?
The integral of sqrt(x)(x^3-1/x) is 2/9 x^4sqrt(x)-2sqrt(x)+C