∫1(β x)(x ∞)dx = ∫1(x α)(x β)dx = ∫(1(β α)(x β) 1(β x)(x α))dx = [1β α∫1x βdx 1β α∫1x αdx] Put x β =u #160; #160; #160; ...

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Definition of Function

∫ dx / x-α β ...

Question

$\int \frac{d x}{(x - α) (β - x)}$

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Solution

$\int \frac{1}{(β - x) (x - \infty)} d x$
$= - \int \frac{1}{(x - α) (x - β)} d x$
$= - \int (\frac{1}{(β - α) (x - β)} - \frac{1}{(β - x) (x - α)}) d x$
$= - [\frac{1}{β - α} \int \frac{1}{x - β} d x - \frac{1}{β - α} \int \frac{1}{x - α} d x]$
Put $x - β = u$ & Put $x - \infty = v$
$d x = d u$ $d x = d v$
$= - [\frac{1}{β - α} \int \frac{1}{u} d u - \frac{1}{β - α} \int \frac{1}{v} d v]$
$= - [\frac{1}{β - α} l n (u) - \frac{1}{β - α} l n (v)]$
$= - [\frac{l n (x - β)}{β - α} - \frac{l n (x - α}{β - α}]$ .

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