The correct option is C log |(x 2)^2x 1 |+C Let x/(x 1)(x 2)=A/(x 1)+B/(x 2) ⇒ x=A(x 2)+B(x 1) ............ (1)Substituting x=1 and 2 in ...

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Definite Integral as Limit of Sum

∫xdx/x-1x-2 e...

Question

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

log ∣ ∣ ∣ \frac{(x - 1)^{2}}{x - 2} ∣ ∣ ∣ + C

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log ∣ ∣ ∣ \frac{(x - 2)^{2}}{x - 1} ∣ ∣ ∣ + C

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log ∣ ∣ ∣ {(\frac{x - 1}{x - 2})}^{2} ∣ ∣ ∣ + C

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log | (x - 1) (x - 2) | + C

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Solution

The correct option is C $log ∣ ∣ ∣ \frac{(x - 2)^{2}}{x - 1} ∣ ∣ ∣ + C$
Let $\frac{x}{(x - 1) (x - 2)} = \frac{A}{(x - 1)} + \frac{B}{(x - 2)}$
$\Rightarrow x = A (x - 2) + B (x - 1)$ ............ (1)
Substituting $x = 1$ and 2 in (1), we obtain
$A = - 1$ and $B = 2$
$∴ \frac{x}{(x - 1) (x - 2)} = - \frac{1}{(x - 1)} + \frac{2}{(x - 2)}$
$\Rightarrow \int \frac{x}{(x - 1) (x - 2)} d x = \int {\frac{- 1}{(x - 1)} + \frac{2}{(x - 2)}} d x$
$= - log | x - 1 | + 2 log | x - 2 | + C$
$= log ∣ ∣ ∣ \frac{(x - 2)^{2}}{x - 1} ∣ ∣ ∣ + C$

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