Evaluate ∫ (3x - 1)/((x - 2)^2) dx.

Question 72

HardOriginal question

Evaluate $\int \frac{3x - 1}{(x - 2)^{2}}\,dx$ .

3\log\left|x - 2\right| + \frac{5}{x - 2} + C

Correct answerYour answer

3\log\left|x - 2\right| - \frac{5}{x - 2} + C

Correct answerYour answer

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

Correct answerYour answer

3\log\left|x - 2\right| - \frac{1}{x - 2} + C

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

$\frac{3x - 1}{(x - 2)^{2}} = \frac{A}{x - 2} + \frac{B}{(x - 2)^{2}}$ : $3x - 1 = A(x - 2) + B$ , so $A = 3$ and, at $x = 2$ , $B = 5$ .
$\int \frac{5}{(x - 2)^{2}}\,dx = -\frac{5}{x - 2}$ , so the integral is $3\log|x - 2| - \frac{5}{x - 2} + C$ .

Worth noting

$+\frac{5}{x - 2}$ loses the sign: $\int (x - 2)^{-2}\,dx = -(x - 2)^{-1}$ .

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