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

The integral of ((x-3)^3)/(x^2-9) is (x^2)/2-9x+36ln|x+3|+C

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

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

\frac{x ^{2}}{2} - 9 x + 36 ln ∣ x + 3 ∣ + C

Solution steps

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

Long division

\frac{( x - 3 ) ^{3}}{x ^{2} - 9} : x - 9 + \frac{36}{x + 3}

= \int x - 9 + \frac{36}{x + 3} 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 x d x - \int 9 d x + \int \frac{36}{x + 3} d x

\int x d x = \frac{x ^{2}}{2}

\int 9 d x = 9 x

\int \frac{36}{x + 3} d x = 36 ln ∣ x + 3 ∣

= \frac{x ^{2}}{2} - 9 x + 36 ln ∣ x + 3 ∣

Add a constant to the solution

= \frac{x ^{2}}{2} - 9 x + 36 ln ∣ x + 3 ∣ + C

\int \frac{x sin ( x )}{cos ^{2} ( x )} d x

\frac{\partial}{\partial x} (x yz ln (x + y))

x \to 4 - lim (8 - 2 x)

x \to - 8 - lim (\frac{- 2 x}{x + 8})

\int π (x + 2)^{2} d x

What is the integral of ((x-3)^3)/(x^2-9) ?
The integral of ((x-3)^3)/(x^2-9) is (x^2)/2-9x+36ln|x+3|+C