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

The integral of ((x-2)^3)/(3x^2) is 1/3 ((x^2)/2-6x+12ln|x|+8/x)+C

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

integral

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

\frac{1}{3} (\frac{x ^{2}}{2} - 6 x + 12 ln ∣ x ∣ + \frac{8}{x}) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \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}}

= \frac{1}{3} \cdot \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

= \frac{1}{3} (\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{1}{3} (\frac{x ^{2}}{2} - 6 x + 12 ln ∣ x ∣ - (- \frac{8}{x}))

Simplify

= \frac{1}{3} (\frac{x ^{2}}{2} - 6 x + 12 ln ∣ x ∣ + \frac{8}{x})

Add a constant to the solution

= \frac{1}{3} (\frac{x ^{2}}{2} - 6 x + 12 ln ∣ x ∣ + \frac{8}{x}) + C

\frac{t ^{3}}{( 1 + t ^{6} )}

\frac{e ^{3} x}{3}

7 x^{2}

y = 12 (x - 1)^{2} (x + 2)

lo g_{10} (x^{2} + 1)

What is the integral of ((x-2)^3)/(3x^2) ?
The integral of ((x-2)^3)/(3x^2) is 1/3 ((x^2)/2-6x+12ln|x|+8/x)+C