What is the integral of (5x-3)/(18x^3-60x^2+50x) ?

The integral of (5x-3)/(18x^3-60x^2+50x) is -3/50 ln|x|+3/50 ln|3x-5|-8/(15(3x-5))+C

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integral of (5x-3)/(18x^3-60x^2+50x)

\int \frac{5 x - 3}{18 x ^{3} - 60 x ^{2} + 50 x} d x

- \frac{3}{50} ln ∣ x ∣ + \frac{3}{50} ln ∣ 3 x - 5 ∣ - \frac{8}{15 ( 3 x - 5 )} + C

Solution steps

\int \frac{5 x - 3}{18 x ^{3} - 60 x ^{2} + 50 x} d x

Take the partial fraction of

\frac{5 x - 3}{18 x ^{3} - 60 x ^{2} + 50 x} : - \frac{3}{50 x} + \frac{9}{50 ( 3 x - 5 )} + \frac{8}{5 ( 3 x - 5 ) ^{2}}

= \int - \frac{3}{50 x} + \frac{9}{50 ( 3 x - 5 )} + \frac{8}{5 ( 3 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}{50 x} d x + \int \frac{9}{50 ( 3 x - 5 )} d x + \int \frac{8}{5 ( 3 x - 5 ) ^{2}} d x

\int \frac{3}{50 x} d x = \frac{3}{50} ln ∣ x ∣

\int \frac{9}{50 ( 3 x - 5 )} d x = \frac{3}{50} ln ∣ 3 x - 5 ∣

\int \frac{8}{5 ( 3 x - 5 ) ^{2}} d x = - \frac{8}{15 ( 3 x - 5 )}

= - \frac{3}{50} ln ∣ x ∣ + \frac{3}{50} ln ∣ 3 x - 5 ∣ - \frac{8}{15 ( 3 x - 5 )}

Add a constant to the solution

= - \frac{3}{50} ln ∣ x ∣ + \frac{3}{50} ln ∣ 3 x - 5 ∣ - \frac{8}{15 ( 3 x - 5 )} + C

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What is the integral of (5x-3)/(18x^3-60x^2+50x) ?
The integral of (5x-3)/(18x^3-60x^2+50x) is -3/50 ln|x|+3/50 ln|3x-5|-8/(15(3x-5))+C