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

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

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

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

- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣ + C

Solution steps

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

Take the partial fraction of

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

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

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

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

= - ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣

Add a constant to the solution

= - ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣ + C

\int_{- 3}^{3} \frac{18}{x ^{2} - 6 x - 72} d x

\frac{\partial}{\partial y} (\frac{2 x}{( 2 x + 3 y ) ^{2}})

f (x) = e^{t}

y = 2 x + 1, (4, 3)

\int \frac{x ^{2}}{e ^{\frac{x}{3}}} d x

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