Evaluate ∫ (x + 1/x)^2 dx.

Question 11

MediumOriginal question

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

\frac{1}{3}\left(x + \frac{1}{x}\right)^{3} + C

Correct answerYour answer

\frac{x^{3}}{3} + 2x + \frac{1}{x} + C

Correct answerYour answer

\frac{x^{3}}{3} - \frac{1}{x} + C

Correct answerYour answer

\frac{x^{3}}{3} + 2x - \frac{1}{x} + C

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

Expand first: $\left(x + \frac{1}{x}\right)^{2} = x^{2} + 2 + \frac{1}{x^{2}}$ .
$\int \left(x^{2} + 2 + x^{-2}\right)dx = \frac{x^{3}}{3} + 2x - \frac{1}{x} + C$ .

Worth noting

$\frac{1}{3}\left(x + \frac{1}{x}\right)^{3}$ treats the bracket like $x$ ; its derivative carries an extra factor $1 - \frac{1}{x^{2}}$ .

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