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Definite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals3 Marks
Question
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\log\tan\text{x dx}$
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Answer

Let, $\text{I}=\int\limits^{\frac{\pi}{2}}_0\log\tan\text{x dx}\ ....(\text{i})$

$=\int\limits^{\frac{\pi}{2}}_0\log\tan\Big(\frac{\pi}{2}-\text{x}\Big)\text{dx}$ $\Big[\text{Using},\int\limits^{\text{a}}_0\text{f(x)}\text{dx}=\int\limits^{\text{a}}_0\text{f}(\text{a}-\text{x})\text{dx}\Big]$

$=\int\limits^{\frac{\pi}{2}}_0\log\cot\text{x dx}\ ...(\text{ii})$

Adding (i) and (ii) we get

$2\text{I}=\int\limits^{\frac{\pi}{2}}_0\log\tan\text{x dx}+\int\limits^{\frac{\pi}{2}}_0\log\cot\text{x dx}$

$=\int\limits^{\frac{\pi}{2}}_0\log(\tan\text{x}\cdot\cot\text{x})\text{ dx}$

$=\int\limits^{\frac{\pi}{2}}_0\log1\text{ dx}=0$

Hence, $\text{I}=0$ 

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