What is the integral of ((x^3-8))/((x-2)) ?

The integral of ((x^3-8))/((x-2)) is (x^3)/3+x^2+4x+C

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integral of ((x^3-8))/((x-2))

\int \frac{( x ^{3} - 8 )}{( x - 2 )} d x

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

Solution steps

\int \frac{x ^{3} - 8}{x - 2} d x

Simplify

\frac{x ^{3} - 8}{x - 2} : x^{2} + 2 x + 4

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

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

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

\int 4 d x = 4 x

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

Add a constant to the solution

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

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What is the integral of ((x^3-8))/((x-2)) ?
The integral of ((x^3-8))/((x-2)) is (x^3)/3+x^2+4x+C