Evaluate ∫ (cos 2x)/((cos x + sin x)^2) dx.

Question 42

HardOriginal question

Evaluate $\int \frac{\cos 2x}{(\cos x + \sin x)^2}\,dx$ .

\log\left|\cos x + \sin x\right| + C

Correct answerYour answer

\log\left|\cos x - \sin x\right| + C

Correct answerYour answer

-\frac{1}{\cos x + \sin x} + C

Correct answerYour answer

\frac{\sin 2x}{2} + C

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

$\cos 2x = \cos^2 x - \sin^2 x = (\cos x - \sin x)(\cos x + \sin x)$ , so the integrand is $\frac{\cos x - \sin x}{\cos x + \sin x}$ .
The numerator is the derivative of the denominator, so the integral is $\log|\cos x + \sin x| + C$ .

Worth noting

$\log|\cos x - \sin x|$ puts the wrong factor in the logarithm: it is the denominator that remains.

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