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

The integral of ((x^2-x+30))/(x^3+5x) is 6ln|x|-5/2 ln|x^2+5|-1/(sqrt(5))arctan(x/(sqrt(5)))+C

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

\int \frac{( x ^{2} - x + 30 )}{x ^{3} + 5 x} d x

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

Solution steps

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

Take the partial fraction of

\frac{x ^{2} - x + 30}{x ^{3} + 5 x} : \frac{6}{x} + \frac{- 5 x - 1}{x ^{2} + 5}

= \int \frac{6}{x} + \frac{- 5 x - 1}{x ^{2} + 5} 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{6}{x} d x + \int \frac{- 5 x - 1}{x ^{2} + 5} d x

\int \frac{6}{x} d x = 6 ln ∣ x ∣

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

= 6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 5 - \frac{1}{5} arctan (\frac{x}{5})

Add a constant to the solution

= 6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 5 - \frac{1}{5} arctan (\frac{x}{5}) + C

3 x + 6 x^{3}

\frac{d}{d x} (2^{x + 1})

\int e^{x} e^{2 x} + 4 d x

\int cos^{4} (x) \cdot sin^{3} (x) d x

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

What is the integral of ((x^2-x+30))/(x^3+5x) ?
The integral of ((x^2-x+30))/(x^3+5x) is 6ln|x|-5/2 ln|x^2+5|-1/(sqrt(5))arctan(x/(sqrt(5)))+C