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

The integral of ((x-2)^3)/((x-2)) is (x^3)/3-2x^2+4x+C

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

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

\frac{x ^{3}}{3} - 2 x^{2} + 4 x + C

Solution steps

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

Cancel the common factor:

x - 2

= \int (x - 2)^{2} d x

Expand

(x - 2)^{2} : x^{2} - 4 x + 4

= \int x^{2} - 4 x + 4 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 4 x d x + \int 4 d x

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

\int 4 x d x = 2 x^{2}

\int 4 d x = 4 x

= \frac{x ^{3}}{3} - 2 x^{2} + 4 x

Add a constant to the solution

= \frac{x ^{3}}{3} - 2 x^{2} + 4 x + C

arcsec (e^{2 x})

\frac{d y}{d x} = 14 x^{8}

\frac{d}{d x} (x \cdot ∣ x ∣)

x \to - 1 lim (x + a)

x \to 2 lim (2 e^{x - 2})

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