What is the integral of (4x)/(x^2+3x+2) ?

The integral of (4x)/(x^2+3x+2) is 4(-ln|x+1|+2ln|x+2|)+C

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integral of (4x)/(x^2+3x+2)

\int \frac{4 x}{x ^{2} + 3 x + 2} d x

4 (- ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣) + C

Solution steps

\int \frac{4 x}{x ^{2} + 3 x + 2} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int \frac{x}{x ^{2} + 3 x + 2} d x

Take the partial fraction of

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

= 4 \cdot \int - \frac{1}{x + 1} + \frac{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

= 4 (- \int \frac{1}{x + 1} d x + \int \frac{2}{x + 2} d x)

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

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

= 4 (- ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣)

Add a constant to the solution

= 4 (- ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣) + C

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What is the integral of (4x)/(x^2+3x+2) ?
The integral of (4x)/(x^2+3x+2) is 4(-ln|x+1|+2ln|x+2|)+C