What is the integral of (5x^2-11x-3)/((x^2+1)(x-6)) ?

The integral of (5x^2-11x-3)/((x^2+1)(x-6)) is ln|x^2+1|+3ln|x-6|+arctan(x)+C

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integral of (5x^2-11x-3)/((x^2+1)(x-6))

\int \frac{5 x ^{2} - 11 x - 3}{( x ^{2} + 1 ) ( x - 6 )} d x

ln x^{2} + 1 + 3 ln ∣ x - 6 ∣ + arctan (x) + C

Solution steps

\int \frac{5 x ^{2} - 11 x - 3}{( x ^{2} + 1 ) ( x - 6 )} d x

Expand

\frac{5 x ^{2} - 11 x - 3}{( x ^{2} + 1 ) ( x - 6 )} : \frac{5 x ^{2}}{( x ^{2} + 1 ) ( x - 6 )} - \frac{11 x}{( x ^{2} + 1 ) ( x - 6 )} - \frac{3}{( x ^{2} + 1 ) ( x - 6 )}

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{5 x ^{2}}{( x ^{2} + 1 ) ( x - 6 )} d x - \int \frac{11 x}{( x ^{2} + 1 ) ( x - 6 )} d x - \int \frac{3}{( x ^{2} + 1 ) ( x - 6 )} d x

\int \frac{5 x ^{2}}{( x ^{2} + 1 ) ( x - 6 )} d x = 5 (\frac{1}{37} (\frac{1}{2} ln x^{2} + 1 + 6 arctan (x)) + \frac{36}{37} ln ∣ x - 6 ∣)

\int \frac{11 x}{( x ^{2} + 1 ) ( x - 6 )} d x = 11 (\frac{1}{37} (- 3 ln x^{2} + 1 + arctan (x)) + \frac{6}{37} ln ∣ x - 6 ∣)

\int \frac{3}{( x ^{2} + 1 ) ( x - 6 )} d x = 3 (\frac{1}{37} (- \frac{1}{2} ln x^{2} + 1 - 6 arctan (x)) + \frac{1}{37} ln ∣ x - 6 ∣)

= 5 (\frac{1}{37} (\frac{1}{2} ln x^{2} + 1 + 6 arctan (x)) + \frac{36}{37} ln ∣ x - 6 ∣) - 11 (\frac{1}{37} (- 3 ln x^{2} + 1 + arctan (x)) + \frac{6}{37} ln ∣ x - 6 ∣) - 3 (\frac{1}{37} (- \frac{1}{2} ln x^{2} + 1 - 6 arctan (x)) + \frac{1}{37} ln ∣ x - 6 ∣)

Simplify

5 (\frac{1}{37} (\frac{1}{2} ln x^{2} + 1 + 6 arctan (x)) + \frac{36}{37} ln ∣ x - 6 ∣) - 11 (\frac{1}{37} (- 3 ln x^{2} + 1 + arctan (x)) + \frac{6}{37} ln ∣ x - 6 ∣) - 3 (\frac{1}{37} (- \frac{1}{2} ln x^{2} + 1 - 6 arctan (x)) + \frac{1}{37} ln ∣ x - 6 ∣) : ln x^{2} + 1 + 3 ln ∣ x - 6 ∣ + arctan (x)

= ln x^{2} + 1 + 3 ln ∣ x - 6 ∣ + arctan (x)

Add a constant to the solution

= ln x^{2} + 1 + 3 ln ∣ x - 6 ∣ + arctan (x) + C

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What is the integral of (5x^2-11x-3)/((x^2+1)(x-6)) ?
The integral of (5x^2-11x-3)/((x^2+1)(x-6)) is ln|x^2+1|+3ln|x-6|+arctan(x)+C