By using the properties of definite integral, evaluate the integr… - Vidyadip

Question

By using the properties of definite integral, evaluate the integral in Exercise:
$\int^{\frac{\pi}{4}}_{0}\log(1+\tan\text{x})\text{dx}$

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Answer

$\text{Let}\ \text{I}=\int\limits_{0}^{\frac{\pi}{4}}\log(1+\tan\text{x})\text{dx}$$\Rightarrow\ \ \text{I}=\int^{\frac{\pi}{4}}\limits_{0}\log\bigg[1+\tan\bigg(\frac{\pi}{4}-\text{x}\bigg)\bigg]\text{dx}\ \ \bigg[\because\int\limits_{0}^{\text{a}}\text{f}\text{(x)}\ \text{dx}=\int\limits_{0}^{\text{a}}\text{f}\text{(a}-\text{x})\text{dx}=\bigg]$
$\Rightarrow\ \ \text{I}=\int\limits_{0}^{\frac{\pi}{4}}\log\bigg[1+\frac{1-\tan\text{x}}{1+\tan\text{x}}\bigg]\text{dx}=\int\limits_{0}^{\frac{\pi}{4}}\log\bigg[\frac{1+\tan\text{x}+1-\tan\text{x}}{1+\tan\text{x}}\bigg]\text{dx}=\int\limits_{0}^{\frac{\pi}{4}}\log\bigg[\frac{2}{1+\tan\text{x}}\bigg]\text{dx}$
Adding eq. (i) and (ii),
$21=\int\limits_{0}^{\frac{\pi}{4}}\bigg[\log(1+\tan\text{x})+\log\bigg(\frac{2}{1+\tan\text{x}}\bigg)\bigg]\text{dx}=\int\limits_{0 }^{\frac{\pi}{4}}\bigg[\log(1+\tan\text{x)}\bigg(\frac{2}{1+\tan\text{x}}\bigg)\bigg]\text{dx}$
$\Rightarrow\ \ \ 21=\int\limits_{0}^{\frac{\pi}{4}}\big[\log2\big]\text{dx}=(\log2)\ \text{(x)}^{\frac{\pi}{4}}_{0}=\frac{\pi}{4}\log2$
$\Rightarrow\ \ \text{I}=\frac{\pi}{8}\log2$

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