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

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

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

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

- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣ + C

Solution steps

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

Take the partial fraction of

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

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f (x) = x^{3} - 4 x + 6

\frac{d}{d x} (\frac{x ^{2}}{1 + e ^{x}})

\int \frac{10 x ^{2}}{x ^{4} - 1} d x

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