What is the integral of (x^2-x+14)/(x^3+7x) ?

The integral of (x^2-x+14)/(x^3+7x) is 2ln|x|-1/2 ln|x^2+7|-1/(sqrt(7))arctan(x/(sqrt(7)))+C

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integral of (x^2-x+14)/(x^3+7x)

\int \frac{x ^{2} - x + 14}{x ^{3} + 7 x} d x

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

Solution steps

\int \frac{x ^{2} - x + 14}{x ^{3} + 7 x} d x

Take the partial fraction of

\frac{x ^{2} - x + 14}{x ^{3} + 7 x} : \frac{2}{x} + \frac{- x - 1}{x ^{2} + 7}

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

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

Add a constant to the solution

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

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

y^{^{''}} - y = t^{2} e^{t}

\int \frac{x ^{2}}{x ^{3}} d x

\frac{d y}{( 6 + y ) ^{2}} = \frac{d x}{( 10 + 9 x ) ^{4}}

What is the integral of (x^2-x+14)/(x^3+7x) ?
The integral of (x^2-x+14)/(x^3+7x) is 2ln|x|-1/2 ln|x^2+7|-1/(sqrt(7))arctan(x/(sqrt(7)))+C