What is the integral of (3x^2-5x+11)/((x-2)(x^2+9)) ?

The integral of (3x^2-5x+11)/((x-2)(x^2+9)) is ln|x-2|+ln|x^2+9|-1/3 arctan(x/3)+C

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integral of (3x^2-5x+11)/((x-2)(x^2+9))

\int \frac{3 x ^{2} - 5 x + 11}{( x - 2 ) ( x ^{2} + 9 )} d x

ln ∣ x - 2 ∣ + ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3}) + C

Solution steps

\int \frac{3 x ^{2} - 5 x + 11}{( x - 2 ) ( x ^{2} + 9 )} d x

Take the partial fraction of

\frac{3 x ^{2} - 5 x + 11}{( x - 2 ) ( x ^{2} + 9 )} : \frac{1}{x - 2} + \frac{2 x - 1}{x ^{2} + 9}

= \int \frac{1}{x - 2} + \frac{2 x - 1}{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{1}{x - 2} d x + \int \frac{2 x - 1}{x ^{2} + 9} d x

\int \frac{1}{x - 2} d x = ln ∣ x - 2 ∣

\int \frac{2 x - 1}{x ^{2} + 9} d x = ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3})

= ln ∣ x - 2 ∣ + ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3})

Add a constant to the solution

= ln ∣ x - 2 ∣ + ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3}) + C

\int \frac{3 x - 2}{2 x ^{2} - 5 x + 2} d x

\int (\frac{6 x - 3}{x}) d x

\int sec (h^{3}) x d x

\int 3^{x^{2}} \cdot 4^{x^{2}} d x

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

What is the integral of (3x^2-5x+11)/((x-2)(x^2+9)) ?
The integral of (3x^2-5x+11)/((x-2)(x^2+9)) is ln|x-2|+ln|x^2+9|-1/3 arctan(x/3)+C