What is the integral of (x^2+4x+27)/((x-2)(x^2+9)) ?

The integral of (x^2+4x+27)/((x-2)(x^2+9)) is 3ln|x-2|-ln|x^2+9|+C

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integral of (x^2+4x+27)/((x-2)(x^2+9))

\int \frac{x ^{2} + 4 x + 27}{( x - 2 ) ( x ^{2} + 9 )} d x

3 ln ∣ x - 2 ∣ - ln x^{2} + 9 + C

Solution steps

\int \frac{x ^{2} + 4 x + 27}{( x - 2 ) ( x ^{2} + 9 )} d x

Take the partial fraction of

\frac{x ^{2} + 4 x + 27}{( x - 2 ) ( x ^{2} + 9 )} : \frac{3}{x - 2} - \frac{2 x}{x ^{2} + 9}

= \int \frac{3}{x - 2} - \frac{2 x}{x ^{2} + 9} 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{3}{x - 2} d x - \int \frac{2 x}{x ^{2} + 9} d x

\int \frac{3}{x - 2} d x = 3 ln ∣ x - 2 ∣

\int \frac{2 x}{x ^{2} + 9} d x = ln x^{2} + 9

= 3 ln ∣ x - 2 ∣ - ln x^{2} + 9

Add a constant to the solution

= 3 ln ∣ x - 2 ∣ - ln x^{2} + 9 + C

\frac{1}{x ^{3}}

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\int \frac{x sin ( x )}{cos ^{3} ( x )} d x

What is the integral of (x^2+4x+27)/((x-2)(x^2+9)) ?
The integral of (x^2+4x+27)/((x-2)(x^2+9)) is 3ln|x-2|-ln|x^2+9|+C