What is the integral of (3x+5)/(x^2-3x+2) ?

The integral of (3x+5)/(x^2-3x+2) is 3/2 ln|4x^2-12x+8|-19/2 ln|2x-2|+19/2 ln|2x-4|+C

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integral of (3x+5)/(x^2-3x+2)

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

\frac{3}{2} ln 4 x^{2} - 12 x + 8 - \frac{19}{2} ln ∣ 2 x - 2 ∣ + \frac{19}{2} ln ∣ 2 x - 4 ∣ + C

Solution steps

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

Expand

\frac{3 x + 5}{x ^{2} - 3 x + 2} : \frac{3 x}{x ^{2} - 3 x + 2} + \frac{5}{x ^{2} - 3 x + 2}

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

\int \frac{3 x}{x ^{2} - 3 x + 2} d x = 6 (\frac{1}{4} ln 4 x^{2} - 12 x + 8 - \frac{3}{4} (ln ∣ 2 x - 2 ∣ - ln ∣ 2 x - 4 ∣))

\int \frac{5}{x ^{2} - 3 x + 2} d x = - 10 (\frac{1}{2} ln ∣ 2 x - 2 ∣ - \frac{1}{2} ln ∣ 2 x - 4 ∣)

= 6 (\frac{1}{4} ln 4 x^{2} - 12 x + 8 - \frac{3}{4} (ln ∣ 2 x - 2 ∣ - ln ∣ 2 x - 4 ∣)) - 10 (\frac{1}{2} ln ∣ 2 x - 2 ∣ - \frac{1}{2} ln ∣ 2 x - 4 ∣)

Simplify

6 (\frac{1}{4} ln 4 x^{2} - 12 x + 8 - \frac{3}{4} (ln ∣ 2 x - 2 ∣ - ln ∣ 2 x - 4 ∣)) - 10 (\frac{1}{2} ln ∣ 2 x - 2 ∣ - \frac{1}{2} ln ∣ 2 x - 4 ∣) : \frac{3}{2} ln 4 x^{2} - 12 x + 8 - \frac{19}{2} ln ∣ 2 x - 2 ∣ + \frac{19}{2} ln ∣ 2 x - 4 ∣

= \frac{3}{2} ln 4 x^{2} - 12 x + 8 - \frac{19}{2} ln ∣ 2 x - 2 ∣ + \frac{19}{2} ln ∣ 2 x - 4 ∣

Add a constant to the solution

= \frac{3}{2} ln 4 x^{2} - 12 x + 8 - \frac{19}{2} ln ∣ 2 x - 2 ∣ + \frac{19}{2} ln ∣ 2 x - 4 ∣ + C

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What is the integral of (3x+5)/(x^2-3x+2) ?
The integral of (3x+5)/(x^2-3x+2) is 3/2 ln|4x^2-12x+8|-19/2 ln|2x-2|+19/2 ln|2x-4|+C