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

The integral of (x^3)/(x-2) is 3x^2+8ln|x-2|+1/3 (x-2)^3-12+C

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

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

3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12 + C

Solution steps

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

Apply u-substitution

= \int \frac{( u + 2 ) ^{3}}{u} d u

Expand

\frac{( u + 2 ) ^{3}}{u} : u^{2} + 6 u + 12 + \frac{8}{u}

= \int u^{2} + 6 u + 12 + \frac{8}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u^{2} d u + \int 6 u d u + \int 12 d u + \int \frac{8}{u} d u

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

\int 6 u d u = 3 u^{2}

\int 12 d u = 12 u

\int \frac{8}{u} d u = 8 ln ∣ u ∣

= \frac{u ^{3}}{3} + 3 u^{2} + 12 u + 8 ln ∣ u ∣

Substitute back

u = x - 2

= \frac{( x - 2 ) ^{3}}{3} + 3 (x - 2)^{2} + 12 (x - 2) + 8 ln ∣ x - 2 ∣

Simplify

\frac{( x - 2 ) ^{3}}{3} + 3 (x - 2)^{2} + 12 (x - 2) + 8 ln ∣ x - 2 ∣ : 3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12

= 3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12

Add a constant to the solution

= 3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12 + C

What is the integral of (x^3)/(x-2) ?
The integral of (x^3)/(x-2) is 3x^2+8ln|x-2|+1/3 (x-2)^3-12+C