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

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

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

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

x + 4 ln ∣ x ∣ - \frac{5}{x} + C

Solution steps

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

Expand

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

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

\int 1 d x = x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

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

= x + 4 ln ∣ x ∣ - \frac{5}{x}

Add a constant to the solution

= x + 4 ln ∣ x ∣ - \frac{5}{x} + C

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