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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

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

Add a constant to the solution

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

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