Logarithm! (6)

Calculus Level 3

$\large \int_0^{{\pi}/{2}} \ln(\tan x) \sin 2x \, dx = \, ?$

The answer is 0.

1 solution

Samara Simha Reddy
Apr 24, 2016

$\large \displaystyle I = \int_0^{\pi/2} \sin 2x . log(\tan x) . dx = \int_0^{\pi/2} \sin (2x) \log (\cot x) .dx\\ \large \displaystyle \text{By Adding, We get }\\ \large \displaystyle 2I = \int_0^{\pi/2} \sin (2x) (\log (\tan x) + \log (\cot x)).dx\\ \large \displaystyle 2I = \int_0^{\pi/2} \sin (2x) \log (\tan x . \cot x).dx = \int_0^{\pi/2} \sin (2x) . (\log 1) .dx\\ \large \displaystyle \left[ \because \log 1 = 0 \right]\\ \large \displaystyle I = \color{#D61F06}{\boxed0}$

Logarithm! (6)

The answer is 0.

1 solution

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