Evaluate ∫ dx/(x(x^2 + 1)).

Question 70

HardOriginal question

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

\log\left|x\right| + \frac{1}{2}\log\left(x^{2} + 1\right) + C

Correct answerYour answer

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

Correct answerYour answer

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

Correct answerYour answer

\log\left|x\right| - \tan^{-1} x + C

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

$\frac{1}{x(x^{2} + 1)} = \frac{A}{x} + \frac{Bx + C}{x^{2} + 1}$ gives $A = 1$ , $B = -1$ , $C = 0$ .
$\int \left(\frac{1}{x} - \frac{x}{x^{2} + 1}\right)dx = \log|x| - \frac{1}{2}\log(x^{2} + 1) + C$ .

Worth noting

$\int \frac{x}{x^{2} + 1}\,dx = \frac{1}{2}\log(x^{2} + 1)$ : the $\frac{1}{2}$ comes from $d(x^{2} + 1) = 2x\,dx$ .

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