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

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

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

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

\frac{8}{11} ln ∣ x - 2 ∣ + \frac{1}{22} ln x^{2} + 7 + 7 \frac{3}{11} arctan (\frac{x}{7}) + \frac{1}{11} ln \frac{x ^{2}}{7} + 1 + \frac{18}{11 7} arctan (\frac{x}{7}) + C

Solution steps

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

Expand

\frac{x ^{2} + 3 x - 2}{( x - 2 ) ( x ^{2} + 7 )} : \frac{x ^{2}}{( x - 2 ) ( x ^{2} + 7 )} + \frac{3 x}{( x - 2 ) ( x ^{2} + 7 )} - \frac{2}{( x - 2 ) ( x ^{2} + 7 )}

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

\int \frac{x ^{2}}{( x - 2 ) ( x ^{2} + 7 )} d x = \frac{4}{11} ln ∣ x - 2 ∣ + \frac{1}{11} (\frac{7}{2} ln x^{2} + 7 + 27 arctan (\frac{x}{7}))

\int \frac{3 x}{( x - 2 ) ( x ^{2} + 7 )} d x = 3 (\frac{2}{11} ln ∣ x - 2 ∣ + \frac{1}{11} (- ln x^{2} + 7 + 7 arctan (\frac{x}{7})))

\int \frac{2}{( x - 2 ) ( x ^{2} + 7 )} d x = 2 (\frac{1}{11} ln ∣ x - 2 ∣ - \frac{1}{22} ln \frac{x ^{2}}{7} + 1 - \frac{2}{11 7} arctan (\frac{x}{7}))

= \frac{4}{11} ln ∣ x - 2 ∣ + \frac{1}{11} (\frac{7}{2} ln x^{2} + 7 + 27 arctan (\frac{x}{7})) + 3 (\frac{2}{11} ln ∣ x - 2 ∣ + \frac{1}{11} (- ln x^{2} + 7 + 7 arctan (\frac{x}{7}))) - 2 (\frac{1}{11} ln ∣ x - 2 ∣ - \frac{1}{22} ln \frac{x ^{2}}{7} + 1 - \frac{2}{11 7} arctan (\frac{x}{7}))

Simplify

\frac{4}{11} ln ∣ x - 2 ∣ + \frac{1}{11} (\frac{7}{2} ln x^{2} + 7 + 27 arctan (\frac{x}{7})) + 3 (\frac{2}{11} ln ∣ x - 2 ∣ + \frac{1}{11} (- ln x^{2} + 7 + 7 arctan (\frac{x}{7}))) - 2 (\frac{1}{11} ln ∣ x - 2 ∣ - \frac{1}{22} ln \frac{x ^{2}}{7} + 1 - \frac{2}{11 7} arctan (\frac{x}{7})) : \frac{8}{11} ln ∣ x - 2 ∣ + \frac{1}{22} ln x^{2} + 7 + 7 \frac{3}{11} arctan (\frac{x}{7}) + \frac{1}{11} ln \frac{x ^{2}}{7} + 1 + \frac{18}{11 7} arctan (\frac{x}{7})

= \frac{8}{11} ln ∣ x - 2 ∣ + \frac{1}{22} ln x^{2} + 7 + 7 \frac{3}{11} arctan (\frac{x}{7}) + \frac{1}{11} ln \frac{x ^{2}}{7} + 1 + \frac{18}{11 7} arctan (\frac{x}{7})

Add a constant to the solution

= \frac{8}{11} ln ∣ x - 2 ∣ + \frac{1}{22} ln x^{2} + 7 + 7 \frac{3}{11} arctan (\frac{x}{7}) + \frac{1}{11} ln \frac{x ^{2}}{7} + 1 + \frac{18}{11 7} arctan (\frac{x}{7}) + C

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