What is the integral of (6x^2+4x-30)/(x^3+5x^2+6x) ?

The integral of (6x^2+4x-30)/(x^3+5x^2+6x) is 7ln|x+2|+4ln|x+3|-5ln|x|+C

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integral of (6x^2+4x-30)/(x^3+5x^2+6x)

\int \frac{6 x ^{2} + 4 x - 30}{x ^{3} + 5 x ^{2} + 6 x} d x

7 ln ∣ x + 2 ∣ + 4 ln ∣ x + 3 ∣ - 5 ln ∣ x ∣ + C

Solution steps

\int \frac{6 x ^{2} + 4 x - 30}{x ^{3} + 5 x ^{2} + 6 x} d x

Expand

\frac{6 x ^{2} + 4 x - 30}{x ^{3} + 5 x ^{2} + 6 x} : \frac{6 x ^{2}}{x ^{3} + 5 x ^{2} + 6 x} + \frac{4 x}{x ^{3} + 5 x ^{2} + 6 x} - \frac{30}{x ^{3} + 5 x ^{2} + 6 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{6 x ^{2}}{x ^{3} + 5 x ^{2} + 6 x} d x + \int \frac{4 x}{x ^{3} + 5 x ^{2} + 6 x} d x - \int \frac{30}{x ^{3} + 5 x ^{2} + 6 x} d x

\int \frac{6 x ^{2}}{x ^{3} + 5 x ^{2} + 6 x} d x = 6 (- 2 ln ∣ x + 2 ∣ + 3 ln ∣ x + 3 ∣)

\int \frac{4 x}{x ^{3} + 5 x ^{2} + 6 x} d x = 4 (ln ∣ x + 2 ∣ - ln ∣ x + 3 ∣)

\int \frac{30}{x ^{3} + 5 x ^{2} + 6 x} d x = 30 (\frac{1}{6} ln ∣ x ∣ - \frac{1}{2} ln ∣ x + 2 ∣ + \frac{1}{3} ln ∣ x + 3 ∣)

= 6 (- 2 ln ∣ x + 2 ∣ + 3 ln ∣ x + 3 ∣) + 4 (ln ∣ x + 2 ∣ - ln ∣ x + 3 ∣) - 30 (\frac{1}{6} ln ∣ x ∣ - \frac{1}{2} ln ∣ x + 2 ∣ + \frac{1}{3} ln ∣ x + 3 ∣)

Simplify

6 (- 2 ln ∣ x + 2 ∣ + 3 ln ∣ x + 3 ∣) + 4 (ln ∣ x + 2 ∣ - ln ∣ x + 3 ∣) - 30 (\frac{1}{6} ln ∣ x ∣ - \frac{1}{2} ln ∣ x + 2 ∣ + \frac{1}{3} ln ∣ x + 3 ∣) : 7 ln ∣ x + 2 ∣ + 4 ln ∣ x + 3 ∣ - 5 ln ∣ x ∣

= 7 ln ∣ x + 2 ∣ + 4 ln ∣ x + 3 ∣ - 5 ln ∣ x ∣

Add a constant to the solution

= 7 ln ∣ x + 2 ∣ + 4 ln ∣ x + 3 ∣ - 5 ln ∣ x ∣ + C

\int_{3}^{4} \frac{1}{x ^{2} - 4} d x

f (x) = 5 x^{2} - 3 x + 7

\int \frac{1}{4 + 2 sin ( x )} d x

e^{- 5 x^{2}}

x \to - 4 lim (15)

What is the integral of (6x^2+4x-30)/(x^3+5x^2+6x) ?
The integral of (6x^2+4x-30)/(x^3+5x^2+6x) is 7ln|x+2|+4ln|x+3|-5ln|x|+C