What is the integral of (x^2)/((x-6)^3) ?

The integral of (x^2)/((x-6)^3) is ln|x-6|-(12)/(x-6)-(18)/((x-6)^2)+C

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integral of (x^2)/((x-6)^3)

\int \frac{x ^{2}}{( x - 6 ) ^{3}} d x

ln ∣ x - 6 ∣ - \frac{12}{x - 6} - \frac{18}{( x - 6 ) ^{2}} + C

Solution steps

\int \frac{x ^{2}}{( x - 6 ) ^{3}} d x

Apply u-substitution

= \int \frac{( u + 6 ) ^{2}}{u ^{3}} d u

Expand

\frac{( u + 6 ) ^{2}}{u ^{3}} : \frac{1}{u} + \frac{12}{u ^{2}} + \frac{36}{u ^{3}}

= \int \frac{1}{u} + \frac{12}{u ^{2}} + \frac{36}{u ^{3}} 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 \frac{1}{u} d u + \int \frac{12}{u ^{2}} d u + \int \frac{36}{u ^{3}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{12}{u ^{2}} d u = - \frac{12}{u}

\int \frac{36}{u ^{3}} d u = - \frac{18}{u ^{2}}

= ln ∣ u ∣ - \frac{12}{u} - \frac{18}{u ^{2}}

Substitute back

u = x - 6

= ln ∣ x - 6 ∣ - \frac{12}{x - 6} - \frac{18}{( x - 6 ) ^{2}}

Add a constant to the solution

= ln ∣ x - 6 ∣ - \frac{12}{x - 6} - \frac{18}{( x - 6 ) ^{2}} + C

What is the integral of (x^2)/((x-6)^3) ?
The integral of (x^2)/((x-6)^3) is ln|x-6|-(12)/(x-6)-(18)/((x-6)^2)+C