What is the integral of (3x-11)/(x^3-5x^2) ?

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

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integral of (3x-11)/(x^3-5x^2)

\int \frac{3 x - 11}{x ^{3} - 5 x ^{2}} d x

- \frac{4}{25} ln ∣ x ∣ - \frac{11}{5 x} + \frac{4}{25} ln ∣ x - 5 ∣ + C

Solution steps

\int \frac{3 x - 11}{x ^{3} - 5 x ^{2}} d x

Take the partial fraction of

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

= \int - \frac{4}{25 x} + \frac{11}{5 x ^{2}} + \frac{4}{25 ( x - 5 )} 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{4}{25 x} d x + \int \frac{11}{5 x ^{2}} d x + \int \frac{4}{25 ( x - 5 )} d x

\int \frac{4}{25 x} d x = \frac{4}{25} ln ∣ x ∣

\int \frac{11}{5 x ^{2}} d x = - \frac{11}{5 x}

\int \frac{4}{25 ( x - 5 )} d x = \frac{4}{25} ln ∣ x - 5 ∣

= - \frac{4}{25} ln ∣ x ∣ - \frac{11}{5 x} + \frac{4}{25} ln ∣ x - 5 ∣

Add a constant to the solution

= - \frac{4}{25} ln ∣ x ∣ - \frac{11}{5 x} + \frac{4}{25} ln ∣ x - 5 ∣ + C

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