First slide

Evaluation of definite integrals

Question

$\int_{- π / 2}^{π / 2} \log (\frac{2 - \sin θ}{2 + \sin θ}) d θ$

Moderate

A

0
B

1
C

2
D

-1

Solution

$\begin{array}{l} f (θ) = \log (\frac{2 - \sin θ}{2 + \sin θ}) \Rightarrow f (- θ) = \log (\frac{2 + \sin θ}{2 - \sin θ}) \\ f (- θ) = - \log (\frac{2 - \sin θ}{2 + \sin θ}) \\ f (- θ) = - f (θ) \\ ∴ f (θ) is odd \\ ∴ \int_{- π / 2}^{π / 2} f (θ) d θ = 0 \end{array}$

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