What is the integral of (5x^2-7x+10)/(x^3+12x^2+36x) ?

The integral of (5x^2-7x+10)/(x^3+12x^2+36x) is 5/18 ln|x|+85/18 ln|x+6|+(116)/(3(x+6))+C

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integral of (5x^2-7x+10)/(x^3+12x^2+36x)

\int \frac{5 x ^{2} - 7 x + 10}{x ^{3} + 12 x ^{2} + 36 x} d x

\frac{5}{18} ln ∣ x ∣ + \frac{85}{18} ln ∣ x + 6 ∣ + \frac{116}{3 ( x + 6 )} + C

Solution steps

\int \frac{5 x ^{2} - 7 x + 10}{x ^{3} + 12 x ^{2} + 36 x} d x

Take the partial fraction of

\frac{5 x ^{2} - 7 x + 10}{x ^{3} + 12 x ^{2} + 36 x} : \frac{5}{18 x} + \frac{85}{18 ( x + 6 )} - \frac{116}{3 ( x + 6 ) ^{2}}

= \int \frac{5}{18 x} + \frac{85}{18 ( x + 6 )} - \frac{116}{3 ( x + 6 ) ^{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{5}{18 x} d x + \int \frac{85}{18 ( x + 6 )} d x - \int \frac{116}{3 ( x + 6 ) ^{2}} d x

\int \frac{5}{18 x} d x = \frac{5}{18} ln ∣ x ∣

\int \frac{85}{18 ( x + 6 )} d x = \frac{85}{18} ln ∣ x + 6 ∣

\int \frac{116}{3 ( x + 6 ) ^{2}} d x = - \frac{116}{3 ( x + 6 )}

= \frac{5}{18} ln ∣ x ∣ + \frac{85}{18} ln ∣ x + 6 ∣ - (- \frac{116}{3 ( x + 6 )})

Simplify

= \frac{5}{18} ln ∣ x ∣ + \frac{85}{18} ln ∣ x + 6 ∣ + \frac{116}{3 ( x + 6 )}

Add a constant to the solution

= \frac{5}{18} ln ∣ x ∣ + \frac{85}{18} ln ∣ x + 6 ∣ + \frac{116}{3 ( x + 6 )} + C

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What is the integral of (5x^2-7x+10)/(x^3+12x^2+36x) ?
The integral of (5x^2-7x+10)/(x^3+12x^2+36x) is 5/18 ln|x|+85/18 ln|x+6|+(116)/(3(x+6))+C