Question
Evaluate the following definite integrals:
$\int_{\frac{\pi}{4}}^\limits{\frac{\pi}{2}}\cot\text{x}\text{ dx}$
$\int_{\frac{\pi}{4}}^\limits{\frac{\pi}{2}}\cot\text{x}\text{ dx}$
✓
Answer
We have,
$\int_{\frac{\pi}{4}}^\limits{\frac{\pi}{2}}\cot\text{x}\text{ dx}$
We know that $\int\cot\text{x dx}=\log(\sin\text{x})$
Now, $\int_{\frac{\pi}{4}}^\limits{\frac{\pi}{2}}\cot\text{x}\text{ dx}$
$=\big[\log(\sin\text{x})\big]^{\frac{\pi}{2}}_\frac{\pi}{4}$
$=\Big[\log\Big(\sin\frac{\pi}{2}\Big)-\log\Big(\sin\frac{\pi}{4}\Big)\Big]$
$=\Big[\log1-\log\frac{1}{\sqrt{2}}\Big]$
$=\big[0-\log\text{2}\big]$
$=\log\sqrt{2}$ $[\because\log\text{a}^{\text{n}}-\text{n}\log\text{a}\big]$
$=\frac{1}{2}\log2$
$\int_{\frac{\pi}{4}}^\limits{\frac{\pi}{2}}\cot\text{x}\text{ dx}$
We know that $\int\cot\text{x dx}=\log(\sin\text{x})$
Now, $\int_{\frac{\pi}{4}}^\limits{\frac{\pi}{2}}\cot\text{x}\text{ dx}$
$=\big[\log(\sin\text{x})\big]^{\frac{\pi}{2}}_\frac{\pi}{4}$
$=\Big[\log\Big(\sin\frac{\pi}{2}\Big)-\log\Big(\sin\frac{\pi}{4}\Big)\Big]$
$=\Big[\log1-\log\frac{1}{\sqrt{2}}\Big]$
$=\big[0-\log\text{2}\big]$
$=\log\sqrt{2}$ $[\because\log\text{a}^{\text{n}}-\text{n}\log\text{a}\big]$
$=\frac{1}{2}\log2$
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