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

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

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

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

- ln ∣ x ∣ + ln x^{2} + x + 1 + C

Solution steps

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

Take the partial fraction of

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= - ln ∣ x ∣ + ln x^{2} + x + 1

Add a constant to the solution

= - ln ∣ x ∣ + ln x^{2} + x + 1 + C

f (x) = x^{2} - 7 x + 5, at x = 4

\int \frac{2 x ^{2} + 6 x - 5 x}{3 x ^{2}} d x

y = - csc (x) - cos (x)

2 y^{^{''}} + 8 y = 0

\int_{- \infty}^{- 1} e^{2 x} d x

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