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

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

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integral of (x^2+5x-1)/(x^3)

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

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

Solution steps

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

Expand

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

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

Simplify

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

Add a constant to the solution

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

\int (- 3 x + 2) e - x d x

x \to 3 lim (\frac{( x ^{2 - 7 x + 12} )}{( 3 - x )})

\int \frac{y}{( y + 1 ) ^{2}} d y

x \to \infty lim (- 2 x e^{- 2 x})

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