Evaluate ∫ (3 x + 1)/(x^2 - 1) dx.

Question 64

MediumOriginal question

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

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

Correct answerYour answer

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

Correct answerYour answer

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

Correct answerYour answer

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

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

$\frac{3x + 1}{(x - 1)(x + 1)} = \frac{A}{x - 1} + \frac{B}{x + 1}$ : $x = 1$ gives $4 = 2A$ , $x = -1$ gives $-2 = -2B$ , so $A = 2$ , $B = 1$ .
$\int \left(\frac{2}{x - 1} + \frac{1}{x + 1}\right)dx = 2\log|x - 1| + \log|x + 1| + C$ .

Worth noting

Swapping $A$ and $B$ puts the 2 on the wrong factor; put each root back in to check.

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