What is the integral of (3x-11)/(x^2-10x+25) ?

The integral of (3x-11)/(x^2-10x+25) is 3ln|x-5|-4/(x-5)+C

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integral of (3x-11)/(x^2-10x+25)

\int \frac{3 x - 11}{x ^{2} - 10 x + 25} d x

3 ln ∣ x - 5 ∣ - \frac{4}{x - 5} + C

Solution steps

\int \frac{3 x - 11}{x ^{2} - 10 x + 25} d x

Take the partial fraction of

\frac{3 x - 11}{x ^{2} - 10 x + 25} : \frac{3}{x - 5} + \frac{4}{( x - 5 ) ^{2}}

= \int \frac{3}{x - 5} + \frac{4}{( x - 5 ) ^{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

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

\int \frac{3}{x - 5} d x = 3 ln ∣ x - 5 ∣

\int \frac{4}{( x - 5 ) ^{2}} d x = - \frac{4}{x - 5}

= 3 ln ∣ x - 5 ∣ - \frac{4}{x - 5}

Add a constant to the solution

= 3 ln ∣ x - 5 ∣ - \frac{4}{x - 5} + C

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What is the integral of (3x-11)/(x^2-10x+25) ?
The integral of (3x-11)/(x^2-10x+25) is 3ln|x-5|-4/(x-5)+C