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

The integral of 1/((x-3)(x+2)) is 1/5 ln|x-3|-1/5 ln|x+2|+C

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integral of 1/((x-3)(x+2))

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

\frac{1}{5} ln ∣ x - 3 ∣ - \frac{1}{5} ln ∣ x + 2 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{1}{( x - 3 ) ( x + 2 )} : \frac{1}{5 ( x - 3 )} - \frac{1}{5 ( x + 2 )}

= \int \frac{1}{5 ( x - 3 )} - \frac{1}{5 ( 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}{5 ( x - 3 )} d x - \int \frac{1}{5 ( x + 2 )} d x

\int \frac{1}{5 ( x - 3 )} d x = \frac{1}{5} ln ∣ x - 3 ∣

\int \frac{1}{5 ( x + 2 )} d x = \frac{1}{5} ln ∣ x + 2 ∣

= \frac{1}{5} ln ∣ x - 3 ∣ - \frac{1}{5} ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{1}{5} ln ∣ x - 3 ∣ - \frac{1}{5} ln ∣ x + 2 ∣ + C

f (x) = x^{2} + 6, (4, 22)

(1, 6), (0.5, 8)

x \to 0 lim ((1 - 2 x)^{\frac{1}{2}})

\frac{150}{s ( s ^{2} + 8 s + 25 )}

\frac{\partial}{\partial x} (\frac{1}{f ( x )})

What is the integral of 1/((x-3)(x+2)) ?
The integral of 1/((x-3)(x+2)) is 1/5 ln|x-3|-1/5 ln|x+2|+C