What is the integral of ((5x^2+sqrt(x)-4)/(x^2)) ?

The integral of ((5x^2+sqrt(x)-4)/(x^2)) is 5x-2/(sqrt(x))+4/x+C

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integral of ((5x^2+sqrt(x)-4)/(x^2))

\int (\frac{5 x ^{2} + x - 4}{x ^{2}}) d x

5 x - \frac{2}{x} + \frac{4}{x} + C

Solution steps

\int \frac{5 x ^{2} + x - 4}{x ^{2}} d x

Expand

\frac{5 x ^{2} + x - 4}{x ^{2}} : 5 + \frac{1}{x ^{\frac{3}{2}}} - \frac{4}{x ^{2}}

= \int 5 + \frac{1}{x ^{\frac{3}{2}}} - \frac{4}{x ^{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 5 d x + \int \frac{1}{x ^{\frac{3}{2}}} d x - \int \frac{4}{x ^{2}} d x

\int 5 d x = 5 x

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

\int \frac{4}{x ^{2}} d x = - \frac{4}{x}

= 5 x - \frac{2}{x} - (- \frac{4}{x})

Simplify

= 5 x - \frac{2}{x} + \frac{4}{x}

Add a constant to the solution

= 5 x - \frac{2}{x} + \frac{4}{x} + C

What is the integral of ((5x^2+sqrt(x)-4)/(x^2)) ?
The integral of ((5x^2+sqrt(x)-4)/(x^2)) is 5x-2/(sqrt(x))+4/x+C