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

The integral of 2/((x-2)^3) is -1/((x-2)^2)+C

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

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

- \frac{1}{( x - 2 ) ^{2}} + C

Solution steps

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

Apply u-substitution

= 2 \cdot \int \frac{1}{u ^{3}} d u

Apply the Power Rule

= 2 (- \frac{1}{2 u ^{2}})

Substitute back

u = x - 2

= 2 (- \frac{1}{2 ( x - 2 ) ^{2}})

Simplify

2 (- \frac{1}{2 ( x - 2 ) ^{2}}) : - \frac{1}{( x - 2 ) ^{2}}

= - \frac{1}{( x - 2 ) ^{2}}

Add a constant to the solution

= - \frac{1}{( x - 2 ) ^{2}} + C

What is the integral of 2/((x-2)^3) ?
The integral of 2/((x-2)^3) is -1/((x-2)^2)+C