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

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

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

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

ln ∣ x ∣ + \frac{3}{x} - \frac{5}{2 x ^{2}} + C

Solution steps

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

Expand

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

= ln ∣ x ∣ - (- \frac{3}{x}) - \frac{5}{2 x ^{2}}

Simplify

= ln ∣ x ∣ + \frac{3}{x} - \frac{5}{2 x ^{2}}

Add a constant to the solution

= ln ∣ x ∣ + \frac{3}{x} - \frac{5}{2 x ^{2}} + C

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