What is the integral of (9x^3-x^2+45x+5)/(x^3+5x) ?

The integral of (9x^3-x^2+45x+5)/(x^3+5x) is 9x-ln|x^2+5|+ln|x|+C

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integral of (9x^3-x^2+45x+5)/(x^3+5x)

\int \frac{9 x ^{3} - x ^{2} + 45 x + 5}{x ^{3} + 5 x} d x

9 x - ln x^{2} + 5 + ln ∣ x ∣ + C

Solution steps

\int \frac{9 x ^{3} - x ^{2} + 45 x + 5}{x ^{3} + 5 x} d x

Expand

\frac{9 x ^{3} - x ^{2} + 45 x + 5}{x ^{3} + 5 x} : \frac{9 x ^{3}}{x ^{3} + 5 x} - \frac{x ^{2}}{x ^{3} + 5 x} + \frac{45 x}{x ^{3} + 5 x} + \frac{5}{x ^{3} + 5 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{9 x ^{3}}{x ^{3} + 5 x} d x - \int \frac{x ^{2}}{x ^{3} + 5 x} d x + \int \frac{45 x}{x ^{3} + 5 x} d x + \int \frac{5}{x ^{3} + 5 x} d x

\int \frac{9 x ^{3}}{x ^{3} + 5 x} d x = - 95 arctan (\frac{x}{5}) + 9 x

\int \frac{x ^{2}}{x ^{3} + 5 x} d x = \frac{1}{2} ln x^{2} + 5

\int \frac{45 x}{x ^{3} + 5 x} d x = 95 arctan (\frac{x}{5})

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

= - 95 arctan (\frac{x}{5}) + 9 x - \frac{1}{2} ln x^{2} + 5 + 95 arctan (\frac{x}{5}) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 5

Simplify

- 95 arctan (\frac{x}{5}) + 9 x - \frac{1}{2} ln x^{2} + 5 + 95 arctan (\frac{x}{5}) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 5 : 9 x - ln x^{2} + 5 + ln ∣ x ∣

= 9 x - ln x^{2} + 5 + ln ∣ x ∣

Add a constant to the solution

= 9 x - ln x^{2} + 5 + ln ∣ x ∣ + C

x \to 2 lim (x^{4} - 16)

\int 5 t e^{t} d t

x \to 6 - lim (\frac{x + 1}{x - 6})

\frac{d}{d x} (\frac{4 x ^{2} - 3}{x + 4})

\frac{d}{d x} ((2 x + 8)^{2})

What is the integral of (9x^3-x^2+45x+5)/(x^3+5x) ?
The integral of (9x^3-x^2+45x+5)/(x^3+5x) is 9x-ln|x^2+5|+ln|x|+C