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

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

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

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

2 ln ∣ x + 2 ∣ - ln ∣ x + 1 ∣ + C

Solution steps

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

Take the partial fraction of

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

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

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

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

= 2 ln ∣ x + 2 ∣ - ln ∣ x + 1 ∣

Add a constant to the solution

= 2 ln ∣ x + 2 ∣ - ln ∣ x + 1 ∣ + C

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