What is the integral of ((x-4)^3)/(x-3) ?

The integral of ((x-4)^3)/(x-3) is (x^3)/3-(9x^2)/2+21x-ln|x-3|+C

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integral of ((x-4)^3)/(x-3)

\int \frac{( x - 4 ) ^{3}}{x - 3} d x

\frac{x ^{3}}{3} - \frac{9 x ^{2}}{2} + 21 x - ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{( x - 4 ) ^{3}}{x - 3} d x

Long division

\frac{( x - 4 ) ^{3}}{x - 3} : x^{2} - 9 x + 21 - \frac{1}{x - 3}

= \int x^{2} - 9 x + 21 - \frac{1}{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^{2} d x - \int 9 x d x + \int 21 d x - \int \frac{1}{x - 3} d x

\int x^{2} d x = \frac{x ^{3}}{3}

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

\int 21 d x = 21 x

\int \frac{1}{x - 3} d x = ln ∣ x - 3 ∣

= \frac{x ^{3}}{3} - \frac{9 x ^{2}}{2} + 21 x - ln ∣ x - 3 ∣

Add a constant to the solution

= \frac{x ^{3}}{3} - \frac{9 x ^{2}}{2} + 21 x - ln ∣ x - 3 ∣ + C

What is the integral of ((x-4)^3)/(x-3) ?
The integral of ((x-4)^3)/(x-3) is (x^3)/3-(9x^2)/2+21x-ln|x-3|+C