What is the integral of (4x^2-x+25)/(x^3+25x) ?

The integral of (4x^2-x+25)/(x^3+25x) is -1/5 arctan(x/5)+3/2 ln|x^2+25|+ln|x|+C

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integral of (4x^2-x+25)/(x^3+25x)

\int \frac{4 x ^{2} - x + 25}{x ^{3} + 25 x} d x

- \frac{1}{5} arctan (\frac{x}{5}) + \frac{3}{2} ln x^{2} + 25 + ln ∣ x ∣ + C

Solution steps

\int \frac{4 x ^{2} - x + 25}{x ^{3} + 25 x} d x

Expand

\frac{4 x ^{2} - x + 25}{x ^{3} + 25 x} : \frac{4 x ^{2}}{x ^{3} + 25 x} - \frac{x}{x ^{3} + 25 x} + \frac{25}{x ^{3} + 25 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{4 x ^{2}}{x ^{3} + 25 x} d x - \int \frac{x}{x ^{3} + 25 x} d x + \int \frac{25}{x ^{3} + 25 x} d x

\int \frac{4 x ^{2}}{x ^{3} + 25 x} d x = 2 ln x^{2} + 25

\int \frac{x}{x ^{3} + 25 x} d x = \frac{1}{5} arctan (\frac{x}{5})

\int \frac{25}{x ^{3} + 25 x} d x = ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 25

= 2 ln x^{2} + 25 - \frac{1}{5} arctan (\frac{x}{5}) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 25

Simplify

2 ln x^{2} + 25 - \frac{1}{5} arctan (\frac{x}{5}) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 25 : - \frac{1}{5} arctan (\frac{x}{5}) + \frac{3}{2} ln x^{2} + 25 + ln ∣ x ∣

= - \frac{1}{5} arctan (\frac{x}{5}) + \frac{3}{2} ln x^{2} + 25 + ln ∣ x ∣

Add a constant to the solution

= - \frac{1}{5} arctan (\frac{x}{5}) + \frac{3}{2} ln x^{2} + 25 + ln ∣ x ∣ + C

What is the integral of (4x^2-x+25)/(x^3+25x) ?
The integral of (4x^2-x+25)/(x^3+25x) is -1/5 arctan(x/5)+3/2 ln|x^2+25|+ln|x|+C