What is the integral of (x^3+2x^2-9x-6)/(x^2-9) ?

The integral of (x^3+2x^2-9x-6)/(x^2-9) is 1/2 x^2+2x-2ln| x/3+1|+2ln| x/3-1|-9/2+C

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integral of (x^3+2x^2-9x-6)/(x^2-9)

\int \frac{x ^{3} + 2 x ^{2} - 9 x - 6}{x ^{2} - 9} d x

\frac{1}{2} x^{2} + 2 x - 2 ln \frac{x}{3} + 1 + 2 ln \frac{x}{3} - 1 - \frac{9}{2} + C

Solution steps

\int \frac{x ^{3} + 2 x ^{2} - 9 x - 6}{x ^{2} - 9} d x

Expand

\frac{x ^{3} + 2 x ^{2} - 9 x - 6}{x ^{2} - 9} : \frac{x ^{3}}{x ^{2} - 9} + \frac{2 x ^{2}}{x ^{2} - 9} - \frac{9 x}{x ^{2} - 9} - \frac{6}{x ^{2} - 9}

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{x ^{3}}{x ^{2} - 9} d x + \int \frac{2 x ^{2}}{x ^{2} - 9} d x - \int \frac{9 x}{x ^{2} - 9} d x - \int \frac{6}{x ^{2} - 9} d x

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

\int \frac{2 x ^{2}}{x ^{2} - 9} d x = - 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3})

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

\int \frac{6}{x ^{2} - 9} d x = - ln \frac{x}{3} + 1 + ln \frac{x}{3} - 1

= \frac{1}{2} (x^{2} - 9 + 9 ln x^{2} - 9) - 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3}) - \frac{9}{2} ln x^{2} - 9 - (- ln \frac{x}{3} + 1 + ln \frac{x}{3} - 1)

Simplify

\frac{1}{2} (x^{2} - 9 + 9 ln x^{2} - 9) - 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3}) - \frac{9}{2} ln x^{2} - 9 - (- ln \frac{x}{3} + 1 + ln \frac{x}{3} - 1) : \frac{1}{2} x^{2} + 2 x - 2 ln \frac{x}{3} + 1 + 2 ln \frac{x}{3} - 1 - \frac{9}{2}

= \frac{1}{2} x^{2} + 2 x - 2 ln \frac{x}{3} + 1 + 2 ln \frac{x}{3} - 1 - \frac{9}{2}

Add a constant to the solution

= \frac{1}{2} x^{2} + 2 x - 2 ln \frac{x}{3} + 1 + 2 ln \frac{x}{3} - 1 - \frac{9}{2} + C

\frac{x ^{2} + 1}{x}

f (x) = \frac{5}{x}, at x = 4

x \to \infty lim (ln (x^{\frac{4}{x}}))

\int \frac{x ^{3} + 2}{x ^{2} - 1} d x

y^{^{'}} - \frac{8}{x} y = \frac{y ^{5}}{x ^{11}}

What is the integral of (x^3+2x^2-9x-6)/(x^2-9) ?
The integral of (x^3+2x^2-9x-6)/(x^2-9) is 1/2 x^2+2x-2ln| x/3+1|+2ln| x/3-1|-9/2+C