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

The integral of (x^3-5x^2-2)/(x^3) is x-5ln|x|+1/(x^2)+C

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

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

x - 5 ln ∣ x ∣ + \frac{1}{x ^{2}} + C

Solution steps

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

Expand

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

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

\int 1 d x = x

\int \frac{5}{x} d x = 5 ln ∣ x ∣

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

= x - 5 ln ∣ x ∣ - (- \frac{1}{x ^{2}})

Simplify

= x - 5 ln ∣ x ∣ + \frac{1}{x ^{2}}

Add a constant to the solution

= x - 5 ln ∣ x ∣ + \frac{1}{x ^{2}} + C

What is the integral of (x^3-5x^2-2)/(x^3) ?
The integral of (x^3-5x^2-2)/(x^3) is x-5ln|x|+1/(x^2)+C