What is the integral of (2x^2-9x+11)/((x-3)^2(x-1)) ?

The integral of (2x^2-9x+11)/((x-3)^2(x-1)) is ln|x-3|-1/(x-3)+ln|x-1|+C

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integral of (2x^2-9x+11)/((x-3)^2(x-1))

\int \frac{2 x ^{2} - 9 x + 11}{( x - 3 ) ^{2} ( x - 1 )} d x

ln ∣ x - 3 ∣ - \frac{1}{x - 3} + ln ∣ x - 1 ∣ + C

Solution steps

\int \frac{2 x ^{2} - 9 x + 11}{( x - 3 ) ^{2} ( x - 1 )} d x

Take the partial fraction of

\frac{2 x ^{2} - 9 x + 11}{( x - 3 ) ^{2} ( x - 1 )} : \frac{1}{x - 3} + \frac{1}{( x - 3 ) ^{2}} + \frac{1}{x - 1}

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

\int \frac{1}{x - 3} d x = ln ∣ x - 3 ∣

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

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

= ln ∣ x - 3 ∣ - \frac{1}{x - 3} + ln ∣ x - 1 ∣

Add a constant to the solution

= ln ∣ x - 3 ∣ - \frac{1}{x - 3} + ln ∣ x - 1 ∣ + C

What is the integral of (2x^2-9x+11)/((x-3)^2(x-1)) ?
The integral of (2x^2-9x+11)/((x-3)^2(x-1)) is ln|x-3|-1/(x-3)+ln|x-1|+C