First slide

Evaluation of definite integrals

Question

The value of $\int_{- π}^{3 π} \log (\sec θ - \tan θ) d θ$ is

Easy

A

1
B

0
C

2
D

none of these

Solution

$\begin{aligned} I = \int_{- π}^{3 π} \log (\sec θ - \tan θ) d θ \\ = \int_{- π}^{3 π} \log (\sec (2 π - θ) - \tan (2 π - θ)) d θ \\ = \int_{- π}^{3 π} \log (\sec θ + \tan θ) d θ . \end{aligned}$
Thus $2 I = \int_{- π}^{3 π} [\log (\sec θ - \tan θ) + \log (\sec θ + \tan θ)] d θ$
$\begin{array}{l} = \int_{- π}^{3 π} \log (\sec^{2} θ - \tan^{2} θ) d θ \\ = \int_{- π}^{3 π} \log 1 d θ = 0 . \end{array}$

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