First slide

Evaluation of definite integrals

Question

If $f (x) = cosec (x + π / 3) cosec (x + π / 6)$ then the value of $\int_{0}^{π / 2} f (x) d x$ is

Moderate

A

2 log 3
B

– 2 log 3
C

log 3
D

1/4

Solution

$\begin{array}{l} I = \int_{0}^{π / 2} cosec (x + π / 3) cosec (x + π / 6) d x \\ = \frac{1}{\sin (π / 6)} \int_{0}^{π / 2} \frac{\sin [(x + π / 3) - (x + π / 6)}{\sin (x + π / 3) \sin (x + π / 6)} d x \end{array}$
$\begin{array}{l} = 2 \int_{0}^{π / 2} [\cot (x + π / 6) - \cot (x + π / 3)] d x \\ {= 2 \log |\frac{\sin (x + π / 6)}{\sin (x + π / 3)}|]}_{0}^{π / 2} \\ = 2 [\log (\frac{\sqrt{3} / 2}{1 / 2}) - \log (\frac{1 / 2}{\sqrt{3} / 3})] \\ = 2 \log 3 \end{array}$

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