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

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

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

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

- \frac{1}{x} + ln ∣ x + 1 ∣ + C

Solution steps

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

Take the partial fraction of

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

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

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

\int \frac{1}{x + 1} d x = ln ∣ x + 1 ∣

= - \frac{1}{x} + ln ∣ x + 1 ∣

Add a constant to the solution

= - \frac{1}{x} + ln ∣ x + 1 ∣ + C

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