What is the integral of ((9x+14))/((x-2)(x^2+4)) ?

The integral of ((9x+14))/((x-2)(x^2+4)) is 4ln|x-2|-2ln|x^2+4|+1/2 arctan(x/2)+C

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integral of ((9x+14))/((x-2)(x^2+4))

\int \frac{( 9 x + 14 )}{( x - 2 ) ( x ^{2} + 4 )} d x

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

Solution steps

\int \frac{9 x + 14}{( x - 2 ) ( x ^{2} + 4 )} d x

Take the partial fraction of

\frac{9 x + 14}{( x - 2 ) ( x ^{2} + 4 )} : \frac{4}{x - 2} + \frac{- 4 x + 1}{x ^{2} + 4}

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

\int \frac{4}{x - 2} d x = 4 ln ∣ x - 2 ∣

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

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

Add a constant to the solution

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

What is the integral of ((9x+14))/((x-2)(x^2+4)) ?
The integral of ((9x+14))/((x-2)(x^2+4)) is 4ln|x-2|-2ln|x^2+4|+1/2 arctan(x/2)+C