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

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

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

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

ln ∣ x ∣ - \frac{4}{x} + \frac{5}{2 x ^{2}} + C

Solution steps

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

Expand

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

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

Simplify

= ln ∣ x ∣ - \frac{4}{x} + \frac{5}{2 x ^{2}}

Add a constant to the solution

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

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

\int x^{\frac{- 3}{4}} d x

\int - x^{2} e^{- 2 x} d x

\int tan^{2} (\frac{2}{3}) x d x

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