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Definite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals5 Marks
Question
Evaluate the following integrals:$\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{1}{1+\cot^{\frac{3}{2}}\text{x}}\text{ dx}$
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Answer

Let $\text{I}=\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{1}{1+\cot^{\frac{3}{2}}\text{x}}\text{ dx}\ ...(\text{i})$ Then,$\text{I}=\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{1}{1+\cot^{\frac{3}{2}}\big(\frac{\pi}{3}+\frac{\pi}{6}-\text{x}\big)}\text{ dx}$ $\Bigg[\int\limits^{\text{b}}_\text{a}\text{f(x)}\text{dx}=\int\limits^{\text{b}}_\text{a}\text{f}(\text{a}+\text{b}-\text{x})\text{dx}\Bigg]$
$=\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{1}{1+\cot^{\frac{3}{2}}\big(\frac{\pi}{2}-\text{x}\big)}\text{ dx}$
$=\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{1}{1+\tan^{\frac{3}{2}}\text{x}}\text{ dx}$
$=\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{\cot^{\frac{3}{2}}\text{x}}{\cot^{\frac{3}{2}}\text{x}+1}\text{ dx}\ ...(\text{ii})$
Adding (i) and (ii)$2\text{I}=\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{1+\cot^{\frac{3}{2}}\text{x}}{1+\cot^{\frac{3}{2}}\text{x}}\text{ dx}$
$\Rightarrow2\text{I}=\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\text{dx}$
$\Rightarrow2\text{I}=\big[\text{x}\big]^{\frac{\pi}{3}}_\frac{\pi}{6}$
$\Rightarrow2\text{I}=\frac{\pi}{3}-\frac{\pi}{6}=\frac{\pi}{6}$
$\Rightarrow\text{I}=\frac{\pi}{12}$

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