2x-1 (x-1)^2 dx - Vidyadip

Question

$\int\frac{2\text{x}-1}{(\text{x}-1)^2}\text{dx}$

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Answer

$\int\bigg[\frac{2\text{x}-1}{(\text{x}-1)^2}\bigg]\text{dx}$
$=\int\bigg[\frac{2\text{x}-2+2-1}{(\text{x}-1)^2}\bigg]\text{dx}$
$=\int\bigg(\frac{2(\text{x}-1)}{(\text{x}-1)^2}+\frac{1}{(\text{x}-1)^2}\bigg)\text{dx}$
$=2\int\frac{\text{dx}}{\text{x}-1}+\int(\text{x}-1)^{-2}\text{dx}$
$=2\text{ ln}|\text{x}-1|+\frac{(\text{x}-1)^{-2+1}}{-2+1}+\text{C}$
$=2\text{ ln}|\text{x}-1|-\frac{1}{\text{x}-1}+\text{C}$

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