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

The integral of ((x-2)^3)/(x^2) is (x^2)/2-6x+12ln|x|+8/x+C

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

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

\frac{x ^{2}}{2} - 6 x + 12 ln ∣ x ∣ + \frac{8}{x} + C

Solution steps

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

Expand

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

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

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

\int 6 d x = 6 x

\int \frac{12}{x} d x = 12 ln ∣ x ∣

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

= \frac{x ^{2}}{2} - 6 x + 12 ln ∣ x ∣ - (- \frac{8}{x})

Simplify

= \frac{x ^{2}}{2} - 6 x + 12 ln ∣ x ∣ + \frac{8}{x}

Add a constant to the solution

= \frac{x ^{2}}{2} - 6 x + 12 ln ∣ x ∣ + \frac{8}{x} + C

x e^{3 x}

\int_{0}^{1} x 4 - x^{2} d x

\int (x^{2} + 3 x - 1)^{4} (2 x + 3) d x

f (x) = 5 csc^{4} (x)

1000 (1 + \frac{r}{12})^{60}

What is the integral of ((x-2)^3)/(x^2) ?
The integral of ((x-2)^3)/(x^2) is (x^2)/2-6x+12ln|x|+8/x+C