What is the integral of (2x^2+7x+6)/(x^3+2x^2+2x) ?

The integral of (2x^2+7x+6)/(x^3+2x^2+2x) is 3ln|x|-1/2 ln|x^2+2x+2|+2arctan(x+1)+C

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integral of (2x^2+7x+6)/(x^3+2x^2+2x)

\int \frac{2 x ^{2} + 7 x + 6}{x ^{3} + 2 x ^{2} + 2 x} d x

3 ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 2 x + 2 + 2 arctan (x + 1) + C

Solution steps

\int \frac{2 x ^{2} + 7 x + 6}{x ^{3} + 2 x ^{2} + 2 x} d x

Take the partial fraction of

\frac{2 x ^{2} + 7 x + 6}{x ^{3} + 2 x ^{2} + 2 x} : \frac{3}{x} + \frac{- x + 1}{x ^{2} + 2 x + 2}

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

= 3 ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 2 x + 2 + 2 arctan (x + 1)

Add a constant to the solution

= 3 ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 2 x + 2 + 2 arctan (x + 1) + C

What is the integral of (2x^2+7x+6)/(x^3+2x^2+2x) ?
The integral of (2x^2+7x+6)/(x^3+2x^2+2x) is 3ln|x|-1/2 ln|x^2+2x+2|+2arctan(x+1)+C