Evaluate ∫ 2x/((x + 1)(x + 3)) dx.

Question 67

MediumOriginal question

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

3\log\left|x + 1\right| - \log\left|x + 3\right| + C

Correct answerYour answer

\log\left|(x + 1)(x + 3)\right| + C

Correct answerYour answer

\log\left|x + 1\right| + 3\log\left|x + 3\right| + C

Correct answerYour answer

3\log\left|x + 3\right| - \log\left|x + 1\right| + C

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

$2x = A(x + 3) + B(x + 1)$ : $x = -1$ gives $-2 = 2A$ , $A = -1$ ; $x = -3$ gives $-6 = -2B$ , $B = 3$ .
$\int \left(-\frac{1}{x + 1} + \frac{3}{x + 3}\right)dx = 3\log|x + 3| - \log|x + 1| + C$ .

Worth noting

Swapping $A$ and $B$ puts the 3 on the wrong factor; put each root back in to check.

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