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

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

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

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

- \frac{x ^{2}}{( x - 2 ) ^{2}} + 2 ln ∣ x - 2 ∣ - \frac{4}{x - 2} + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

\int - \frac{x}{( x - 2 ) ^{2}} d x = - ln ∣ x - 2 ∣ + \frac{2}{x - 2}

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

Simplify

2 (- \frac{x ^{2}}{2 ( x - 2 ) ^{2}} - (- ln ∣ x - 2 ∣ + \frac{2}{x - 2})) : - \frac{x ^{2}}{( x - 2 ) ^{2}} + 2 ln ∣ x - 2 ∣ - \frac{4}{x - 2}

= - \frac{x ^{2}}{( x - 2 ) ^{2}} + 2 ln ∣ x - 2 ∣ - \frac{4}{x - 2}

Add a constant to the solution

= - \frac{x ^{2}}{( x - 2 ) ^{2}} + 2 ln ∣ x - 2 ∣ - \frac{4}{x - 2} + C

\int_{- 1}^{2} 3 x^{2} + 2 x + 1 d x

cos (cos (cos (x)))

x^{4} - 32 x^{2} + 256

\frac{d}{d x} (\frac{x ^{3}}{18})

f (x) = 4 x (sin (x) + cos (x))

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