What is the integral of (x^3-5x^2-sqrt(x))/(x^2) ?

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

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integral of (x^3-5x^2-sqrt(x))/(x^2)

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

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

Solution steps

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

Expand

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

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

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

\int 5 d x = 5 x

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

= \frac{x ^{2}}{2} - 5 x - (- \frac{2}{x})

Simplify

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

Add a constant to the solution

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

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