Evaluate ∫ x/((x - 2) (x - 1)) dx.

Question 65

MediumOriginal question

Evaluate $\int \frac{x}{\left(x - 2\right) \left(x - 1\right)}\,dx$ .

2\log\left|x - 1\right| - \log\left|x - 2\right| + C

Correct answerYour answer

2\log\left|x - 2\right| - \log\left|x - 1\right| + C

Correct answerYour answer

\log\left|(x - 1)(x - 2)\right| + C

Correct answerYour answer

\log\left|x - 1\right| + 2\log\left|x - 2\right| + C

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

$x = A(x - 2) + B(x - 1)$ : $x = 1$ gives $A = -1$ , $x = 2$ gives $B = 2$ .
$\int \left(-\frac{1}{x - 1} + \frac{2}{x - 2}\right)dx = 2\log|x - 2| - \log|x - 1| + C$ .

Worth noting

$\log|(x - 1)(x - 2)|$ would need both constants to be 1.

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