What is the integral of 1/((1-x)(5-x)) ?

The integral of 1/((1-x)(5-x)) is -1/4 (ln|(x-3)/2+1|-ln|(x-3)/2-1|)+C

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integral of 1/((1-x)(5-x))

\int \frac{1}{( 1 - x ) ( 5 - x )} d x

- \frac{1}{4} (ln \frac{x - 3}{2} + 1 - ln \frac{x - 3}{2} - 1) + C

Solution steps

\int \frac{1}{( 1 - x ) ( 5 - x )} d x

Complete the square

(1 - x) (5 - x) : (x - 3)^{2} - 4

= \int \frac{1}{( x - 3 ) ^{2} - 4} d x

Apply u-substitution

: \int \frac{1}{u ^{2} - 4} d u

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

Apply Integral Substitution

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{v ^{2} - 1} d v

Factor

v^{2} - 1 : - (- v^{2} + 1)

= \frac{1}{2} \cdot \int \frac{1}{- ( - v ^{2} + 1 )} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} (- \int \frac{1}{- v ^{2} + 1} d v)

Use the common integral:

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

= \frac{1}{2} (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

Substitute back

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

Simplify

\frac{1}{2} (- (\frac{ln \frac{x - 3}{2} + 1}{2} - \frac{ln \frac{x - 3}{2} - 1}{2})) : - \frac{1}{4} (ln \frac{x - 3}{2} + 1 - ln \frac{x - 3}{2} - 1)

= - \frac{1}{4} (ln \frac{x - 3}{2} + 1 - ln \frac{x - 3}{2} - 1)

Add a constant to the solution

= - \frac{1}{4} (ln \frac{x - 3}{2} + 1 - ln \frac{x - 3}{2} - 1) + C

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What is the integral of 1/((1-x)(5-x)) ?
The integral of 1/((1-x)(5-x)) is -1/4 (ln|(x-3)/2+1|-ln|(x-3)/2-1|)+C