What is the integral of (3x^2+5)/(x^3+9x) ?

The integral of (3x^2+5)/(x^3+9x) is 11/9 ln|x^2+9|+5/9 ln|x|+C

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integral of (3x^2+5)/(x^3+9x)

\int \frac{3 x ^{2} + 5}{x ^{3} + 9 x} d x

\frac{11}{9} ln x^{2} + 9 + \frac{5}{9} ln ∣ x ∣ + C

Solution steps

\int \frac{3 x ^{2} + 5}{x ^{3} + 9 x} d x

Expand

\frac{3 x ^{2} + 5}{x ^{3} + 9 x} : \frac{3 x ^{2}}{x ^{3} + 9 x} + \frac{5}{x ^{3} + 9 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 ^{2}}{x ^{3} + 9 x} d x + \int \frac{5}{x ^{3} + 9 x} d x

\int \frac{3 x ^{2}}{x ^{3} + 9 x} d x = \frac{3}{2} ln x^{2} + 9

\int \frac{5}{x ^{3} + 9 x} d x = 5 (\frac{1}{9} ln ∣ x ∣ - \frac{1}{18} ln x^{2} + 9)

= \frac{3}{2} ln x^{2} + 9 + 5 (\frac{1}{9} ln ∣ x ∣ - \frac{1}{18} ln x^{2} + 9)

Simplify

\frac{3}{2} ln x^{2} + 9 + 5 (\frac{1}{9} ln ∣ x ∣ - \frac{1}{18} ln x^{2} + 9) : \frac{11}{9} ln x^{2} + 9 + \frac{5}{9} ln ∣ x ∣

= \frac{11}{9} ln x^{2} + 9 + \frac{5}{9} ln ∣ x ∣

Add a constant to the solution

= \frac{11}{9} ln x^{2} + 9 + \frac{5}{9} ln ∣ x ∣ + C

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What is the integral of (3x^2+5)/(x^3+9x) ?
The integral of (3x^2+5)/(x^3+9x) is 11/9 ln|x^2+9|+5/9 ln|x|+C