সৃজনশীলঃ ‘দৃশ্যকল্প-১ : g(x)=cot−1⁡xদৃশ্যকল্প-২…’ যশোর বোর্ড ২০২২ – যোগজীকরণ, উচ্চতর গণিত, উচ্চতর গণিত ১ম পত্র (এইচএসসি)

দেওয়া আছে,

$\begin{array}{cc}& g\left(x\right)={\cot }^{-1}x\\ & \therefore {\int }_1^{\sqrt{3}}xg\left(x\right)dx={\int }_1^{\sqrt{3}}x{\cot }^{-1}xdx\\ & =[{\cot }^{-1}x{\int }_{1}xdx-{\left\{\frac{d}{dx}\right(\cot x\left){\int }_{1}xdxdx\right]}_1^{\sqrt{3}}\\ & ={[\frac{x^2}{2}{\cot }^{-1}x+{\int }_1^2\frac{x^2}{2+x^2)}dx]}_1^{\sqrt{3}}\\ & ={[\frac{x^2}{2}{\cot }^{-1}x+\frac{1}{2}{\int }_1^2\frac{1}{1+x^2}dx]}_1^{\sqrt{3}}\\ & ={[\frac{x^2}{2}{\cot }^{-1}x+\frac{1}{2}{\int }_1^{2}1-\frac{1}{1+x^2}dx]}_1^{\sqrt{3}}\end{array}$

$\begin{array}{cc}& ={[\frac{x^2}{2}{\cot }^{-1}x+\frac{x}{2}+\frac{1}{2}{\cot }^{-1}x+c]}_1^{\sqrt{3}}\\ & \therefore {\int }_1^{\sqrt{3}}x{\cot }^{-1}xdx\\ & ={[\frac{x^2}{2}{\cot }^{-1}x+\frac{x}{2}+\frac{1}{2}{\cot }^{-1}x]}_1^{\sqrt{3}}\\ & =\frac{1}{2}{[x^2{\cot }^{-1}x]}_1^{\sqrt{3}}+\frac{1}{2}[x{]}_1^{\sqrt{3}}+\frac{1}{2}{[{\cot }^{-1}x]}_1^{\sqrt{3}}\\ & =\frac{1}{2}[3{\cot }^{-1}\sqrt{3}-{\cot }^{-1}\frac{1}{11}]+\frac{1}{2}[\sqrt{3}-1]\\ & +\frac{1}{2}({\cot }^{-1}\sqrt{3}-{\cot }^{-1}1)\\ & =\frac{1}{2}(3\cdot \frac{\pi }{6}-\frac{\pi }{4})+\frac{1}{2}(\sqrt{3}-1)+\frac{1}{2}(\frac{\pi }{6}-\frac{\pi }{4})\\ & =\frac{\pi }{4}-\frac{\pi }{8}+\frac{1}{2}(\sqrt{3}-1)+\frac{\pi }{12}-\frac{\pi }{8}\\ & =\frac{1}{2}(\sqrt{3}-1)+\frac{\pi }{12}\\ & =\frac{1}{2}(\sqrt{3}-1+\frac{\pi }{6})\end{array}$

ইহাই নির্ণেয় মান।  

দেওয়া আছ,    

 y² = 2x ........ (i)

x = 3y ........... (ii)

(i) নং সমীকরণে x = 3y বসিয়ে,

হলেহলে$\begin{array}{cc}& 1-2,3y\\ & \Rightarrow y^2=6y\\ & \Rightarrow y^2-6y=0\\ & \Rightarrow y(y-6)=0\\ & \therefore y=0,6\\ & y=0\text{ হলে, }x=0\\ & y=6\text{ হলে, }x=18\end{array}$(i) নং ও (ii) রেখার ছেদবিন্দু (0,0) এবং (18,6)

(i) নং হতে পাই,

$\begin{array}{cc}{y}^2& =2x\\ \Rightarrow y& =\sqrt{2}\sqrt{x}=y_1\end{array}$

(ii) নং হতে পাই,

$\begin{array}{cc}x& =3y\\ \Rightarrow y& =\frac{x}{3}=y_2\end{array}$

(i) ও (ii) নং দ্বারা আবদ্ধ ক্ষেত্রের ক্ষেত্রফল,

$\begin{array}{cc}& ={\int }_0^{18}(y_1-y_2)dx\\ & ={\int }_0^{18}(\sqrt{2}\sqrt{x}-\frac{x}{3})dx\\ & =\sqrt{2}{\int }_0^{18}{[\sqrt{x}dx-\frac{1}{3}]}_0^{18}xdx\\ & =\sqrt{2}{\left[\frac{x^2}{3}\right]}_0^{18}-\frac{1}{3}{\left[\frac{x^2}{2}\right]}_0^{18}\\ & =\frac{2\sqrt{2}}{3}(18{)}^{\frac{3}{2}}-0-\frac{1}{6}\times (18{)}^2+0\\ & =\frac{2\sqrt{2}}{3}(18{)}^{\frac{3}{2}}-15\\ & =72-54\end{array}$

= 18 বর্গ একক।

নির্ণেয় আবদ্ধ ক্ষেত্রের ক্ষেত্রফল 18 বর্গ একক।