What is the integral of ((x^3+5x^2-4)/(x^2)) ?

The integral of ((x^3+5x^2-4)/(x^2)) is (x^2)/2+5x+4/x+C

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integral of ((x^3+5x^2-4)/(x^2))

\int (\frac{x ^{3} + 5 x ^{2} - 4}{x ^{2}}) d x

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

Solution steps

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

Expand

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

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

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

\int 5 d x = 5 x

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

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

Simplify

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

Add a constant to the solution

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

What is the integral of ((x^3+5x^2-4)/(x^2)) ?
The integral of ((x^3+5x^2-4)/(x^2)) is (x^2)/2+5x+4/x+C