What is the integral of (12x^2-4x)/((x^2+4)(x+2)) ?

The integral of (12x^2-4x)/((x^2+4)(x+2)) is 5/2 ln|(x^2)/4+1|-7arctan(x/2)+7ln|x+2|+C

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integral of (12x^2-4x)/((x^2+4)(x+2))

\int \frac{12 x ^{2} - 4 x}{( x ^{2} + 4 ) ( x + 2 )} d x

\frac{5}{2} ln \frac{x ^{2}}{4} + 1 - 7 arctan (\frac{x}{2}) + 7 ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{12 x ^{2} - 4 x}{( x ^{2} + 4 ) ( x + 2 )} d x

Expand

\frac{12 x ^{2} - 4 x}{( x ^{2} + 4 ) ( x + 2 )} : \frac{12 x ^{2}}{( x ^{2} + 4 ) ( x + 2 )} - \frac{4 x}{( x ^{2} + 4 ) ( x + 2 )}

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{12 x ^{2}}{( x ^{2} + 4 ) ( x + 2 )} d x - \int \frac{4 x}{( x ^{2} + 4 ) ( x + 2 )} d x

\int \frac{12 x ^{2}}{( x ^{2} + 4 ) ( x + 2 )} d x = 12 (\frac{1}{2} (\frac{1}{2} ln \frac{x ^{2}}{4} + 1 - arctan (\frac{x}{2})) + \frac{1}{2} ln ∣ x + 2 ∣)

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

= 12 (\frac{1}{2} (\frac{1}{2} ln \frac{x ^{2}}{4} + 1 - arctan (\frac{x}{2})) + \frac{1}{2} ln ∣ x + 2 ∣) - (\frac{1}{2} ln \frac{x ^{2}}{4} + 1 + arctan (\frac{x}{2}) - ln ∣ x + 2 ∣)

Simplify

12 (\frac{1}{2} (\frac{1}{2} ln \frac{x ^{2}}{4} + 1 - arctan (\frac{x}{2})) + \frac{1}{2} ln ∣ x + 2 ∣) - (\frac{1}{2} ln \frac{x ^{2}}{4} + 1 + arctan (\frac{x}{2}) - ln ∣ x + 2 ∣) : \frac{5}{2} ln \frac{x ^{2}}{4} + 1 - 7 arctan (\frac{x}{2}) + 7 ln ∣ x + 2 ∣

= \frac{5}{2} ln \frac{x ^{2}}{4} + 1 - 7 arctan (\frac{x}{2}) + 7 ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{5}{2} ln \frac{x ^{2}}{4} + 1 - 7 arctan (\frac{x}{2}) + 7 ln ∣ x + 2 ∣ + C

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What is the integral of (12x^2-4x)/((x^2+4)(x+2)) ?
The integral of (12x^2-4x)/((x^2+4)(x+2)) is 5/2 ln|(x^2)/4+1|-7arctan(x/2)+7ln|x+2|+C