First slide

Evaluation of definite integrals

Question

$Evaluate \int_{- π}^{3 π} \log (\sec θ - \tan θ) d θ .$

Moderate

A

0
B

2
C

3
D

4

Solution

$Let I = \int_{- π}^{3 π} \log (\sec θ - \tan θ) d θ$ ----------(1)
$Using the property \int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x, we get$
$I = \int_{- π}^{3 π} \log [\sec (2 π - θ) - \tan (2 π - θ)] d θ$
$= \int_{- π}^{3 π} \log [\sec θ + \tan θ] d θ - - - - - - (2)$
$Adding equations (1) and (2), we get$
$\begin{array}{l} 2 I = \int_{- π}^{3 π} \log [(\sec θ - \tan θ) (\sec θ + \tan θ)] d θ \\ = \int_{- π}^{3 π} \log (1) d θ = \int_{- π}^{3 π} 0 . d θ = 0 \\ \Rightarrow I = 0 \end{array}$

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