What is the integral of (x^2-x+28)/(x^3+7x) ?

The integral of (x^2-x+28)/(x^3+7x) is 4ln|x|-3/2 ln|x^2+7|-1/(sqrt(7))arctan(x/(sqrt(7)))+C

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integral of (x^2-x+28)/(x^3+7x)

\int \frac{x ^{2} - x + 28}{x ^{3} + 7 x} d x

4 ln ∣ x ∣ - \frac{3}{2} ln x^{2} + 7 - \frac{1}{7} arctan (\frac{x}{7}) + C

Solution steps

\int \frac{x ^{2} - x + 28}{x ^{3} + 7 x} d x

Take the partial fraction of

\frac{x ^{2} - x + 28}{x ^{3} + 7 x} : \frac{4}{x} + \frac{- 3 x - 1}{x ^{2} + 7}

= \int \frac{4}{x} + \frac{- 3 x - 1}{x ^{2} + 7} 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{4}{x} d x + \int \frac{- 3 x - 1}{x ^{2} + 7} d x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

\int \frac{- 3 x - 1}{x ^{2} + 7} d x = - \frac{3}{2} ln x^{2} + 7 - \frac{1}{7} arctan (\frac{x}{7})

= 4 ln ∣ x ∣ - \frac{3}{2} ln x^{2} + 7 - \frac{1}{7} arctan (\frac{x}{7})

Add a constant to the solution

= 4 ln ∣ x ∣ - \frac{3}{2} ln x^{2} + 7 - \frac{1}{7} arctan (\frac{x}{7}) + C

What is the integral of (x^2-x+28)/(x^3+7x) ?
The integral of (x^2-x+28)/(x^3+7x) is 4ln|x|-3/2 ln|x^2+7|-1/(sqrt(7))arctan(x/(sqrt(7)))+C