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

The integral of (2x)/((x-2)^2(x+2)) is 2(-1/(x-2)-2(1/16 ln|x+2|-1/16 ln|x-2|-1/(4(x-2))))+C

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

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

2 (- \frac{1}{x - 2} - 2 (\frac{1}{16} ln ∣ x + 2 ∣ - \frac{1}{16} ln ∣ x - 2 ∣ - \frac{1}{4 ( x - 2 )})) + C

Solution steps

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

Apply u-substitution

: \int \frac{u - 2}{u ( u - 4 ) ^{2}} d u

= 2 \cdot \int \frac{u - 2}{u ( u - 4 ) ^{2}} d u

Expand

\frac{u - 2}{u ( u - 4 ) ^{2}} : \frac{u}{u ( u - 4 ) ^{2}} - \frac{2}{u ( u - 4 ) ^{2}}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int \frac{u}{u ( u - 4 ) ^{2}} d u - \int \frac{2}{u ( u - 4 ) ^{2}} d u)

\int \frac{u}{u ( u - 4 ) ^{2}} d u = - \frac{1}{u - 4}

\int \frac{2}{u ( u - 4 ) ^{2}} d u = 2 (\frac{1}{16} ln ∣ u ∣ - \frac{1}{16} ln ∣ u - 4 ∣ - \frac{1}{4 ( u - 4 )})

= 2 (- \frac{1}{u - 4} - 2 (\frac{1}{16} ln ∣ u ∣ - \frac{1}{16} ln ∣ u - 4 ∣ - \frac{1}{4 ( u - 4 )}))

Substitute back

u = x + 2

= 2 (- \frac{1}{x + 2 - 4} - 2 (\frac{1}{16} ln ∣ x + 2 ∣ - \frac{1}{16} ln ∣ x + 2 - 4 ∣ - \frac{1}{4 ( x + 2 - 4 )}))

Simplify

2 (- \frac{1}{x + 2 - 4} - 2 (\frac{1}{16} ln ∣ x + 2 ∣ - \frac{1}{16} ln ∣ x + 2 - 4 ∣ - \frac{1}{4 ( x + 2 - 4 )})) : 2 (- \frac{1}{x - 2} - 2 (\frac{1}{16} ln ∣ x + 2 ∣ - \frac{1}{16} ln ∣ x - 2 ∣ - \frac{1}{4 ( x - 2 )}))

= 2 (- \frac{1}{x - 2} - 2 (\frac{1}{16} ln ∣ x + 2 ∣ - \frac{1}{16} ln ∣ x - 2 ∣ - \frac{1}{4 ( x - 2 )}))

Add a constant to the solution

= 2 (- \frac{1}{x - 2} - 2 (\frac{1}{16} ln ∣ x + 2 ∣ - \frac{1}{16} ln ∣ x - 2 ∣ - \frac{1}{4 ( x - 2 )})) + C

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What is the integral of (2x)/((x-2)^2(x+2)) ?
The integral of (2x)/((x-2)^2(x+2)) is 2(-1/(x-2)-2(1/16 ln|x+2|-1/16 ln|x-2|-1/(4(x-2))))+C