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

The integral of (x^2+3x+1)/((x-1)(x-2)(x-3)) is 5/2 ln|x-1|+19/2 ln|x-3|-11ln|x-2|+C

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

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

\frac{5}{2} ln ∣ x - 1 ∣ + \frac{19}{2} ln ∣ x - 3 ∣ - 11 ln ∣ x - 2 ∣ + C

Solution steps

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

Expand

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

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

\int \frac{x ^{2}}{( x - 1 ) ( x - 2 ) ( x - 3 )} d x = \frac{1}{2} ln ∣ x - 1 ∣ - 4 ln ∣ x - 2 ∣ + \frac{9}{2} ln ∣ x - 3 ∣

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

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

= \frac{1}{2} ln ∣ x - 1 ∣ - 4 ln ∣ x - 2 ∣ + \frac{9}{2} ln ∣ x - 3 ∣ + 3 (\frac{1}{2} ln ∣ x - 1 ∣ + \frac{3}{2} ln ∣ x - 3 ∣ - 2 ln ∣ x - 2 ∣) + \frac{1}{2} ln ∣ x - 1 ∣ - ln ∣ x - 2 ∣ + \frac{1}{2} ln ∣ x - 3 ∣

Simplify

\frac{1}{2} ln ∣ x - 1 ∣ - 4 ln ∣ x - 2 ∣ + \frac{9}{2} ln ∣ x - 3 ∣ + 3 (\frac{1}{2} ln ∣ x - 1 ∣ + \frac{3}{2} ln ∣ x - 3 ∣ - 2 ln ∣ x - 2 ∣) + \frac{1}{2} ln ∣ x - 1 ∣ - ln ∣ x - 2 ∣ + \frac{1}{2} ln ∣ x - 3 ∣ : \frac{5}{2} ln ∣ x - 1 ∣ + \frac{19}{2} ln ∣ x - 3 ∣ - 11 ln ∣ x - 2 ∣

= \frac{5}{2} ln ∣ x - 1 ∣ + \frac{19}{2} ln ∣ x - 3 ∣ - 11 ln ∣ x - 2 ∣

Add a constant to the solution

= \frac{5}{2} ln ∣ x - 1 ∣ + \frac{19}{2} ln ∣ x - 3 ∣ - 11 ln ∣ x - 2 ∣ + C

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\int (x^{3} + 1) ln (x) d x

\int x^{3} a^{2} + x^{2} d x

What is the integral of (x^2+3x+1)/((x-1)(x-2)(x-3)) ?
The integral of (x^2+3x+1)/((x-1)(x-2)(x-3)) is 5/2 ln|x-1|+19/2 ln|x-3|-11ln|x-2|+C