What is the integral of 1/((x-2)(x+2)) ?

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

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integral of 1/((x-2)(x+2))

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

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

Solution steps

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

Take the partial fraction of

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

= \int \frac{1}{4 ( x - 2 )} - \frac{1}{4 ( 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

= \int \frac{1}{4 ( x - 2 )} d x - \int \frac{1}{4 ( x + 2 )} d x

\int \frac{1}{4 ( x - 2 )} d x = \frac{1}{4} ln ∣ x - 2 ∣

\int \frac{1}{4 ( x + 2 )} d x = \frac{1}{4} ln ∣ x + 2 ∣

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

Add a constant to the solution

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

x \to 0 lim (\frac{100000}{x})

\int x^{6} ln (3 x) d x

ln (22 x)

sin (2 x) cos (2 x)

x \to - \infty lim (0 \cdot (x))

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