সৃজনশীলঃ ‘f(x)=xsin−1x এবং g(x)=16−x3…’ ২০২৪ – যোগজীকরণ, উচ্চতর গণিত, উচ্চতর গণিত ১ম পত্র (এইচএসসি)

এখানে,

$f\left(x\right)=x{\sin }^{-1}x$

$\int f\left(x\right)dx$

= $\int x{\sin }^{-1}xdx$

= ${\sin }^{-1}x\int xdx-\int \{\frac{d}{dx}({\sin }^{-1}x)\int xdx\}dx$

= ${\sin }^{-1}x\cdot \frac{x^2}{2}-\int \frac{1}{\sqrt{1-x^2}}\cdot \frac{x^2}{2}dx$

= $\frac{x^2{\sin }^{-1}x}{2}-\frac{1}{2}\int \frac{x^2}{\sqrt{1-x^2}}dx$

= $\frac{x^2{\sin }^{-1}x}{2}-\frac{1}{2}\int \frac{1-(1-x^2)}{\sqrt{1-x^2}}dx$

= $\frac{x^2{\sin }^{-1}x}{2}-\frac{1}{2}\int \frac{1}{\sqrt{1-x^2}}dx+\frac{1}{2}\int \frac{1-x^2}{\sqrt{1-x^2}}dx$

= $\frac{x^2{\sin }^{-\frac{1}{x}}}{2}-\frac{1}{2}\int \frac{dx}{\sqrt{1-x^2}}+\frac{1}{2}\int \sqrt{1-x^2}dx$

= $\frac{x^2{\sin }^{-1}x}{2}-\frac{x}{2}{\sin }^{-1}x+\frac{1}{2}[\frac{x\sqrt{1-x^2}}{2}+\frac{1}{2}{\sin }^{-1}x]+c$

= $\frac{x^2{\sin }^{-1}x}{2}-\frac{1}{2}{\sin }^{-1}x+\frac{x\sqrt{1-x^2}}{\left(011\right)}+\frac{{\sin }^{-1}x}{4}+c$

= $\frac{x^2{\sin }^{-1}x}{2}-\frac{1}{4}{\sin }^{-\frac{1}{x}}+\frac{x\sqrt{1-x^2}}{2}+c$ (Ans)

দেওয়া আছে, $g\left(x\right)=\sqrt{16-x^2}$

ধরি, $x=4\sin \theta$

$\Rightarrow \frac{dx}{d\theta }=4\cos \theta$

$\therefore dx=4\cos \theta d\theta$

$\therefore {\int }_0^{4}g\left(x\right)dx={\int }_0^4\sqrt{16-x^2}dx$

$={\int }_0^{\pi /2}\cdot \sqrt{(4{)}^2-(4\sin \theta {)}^2}\cdot 4\cos \theta d\theta .$

$={\int }_0^{\pi /2}\cdot \sqrt{(4{)}^2-4^2{\sin }^2\theta }\cdot 4\cos \theta d\theta .$

$={\int }_0^{\pi /2}\cdot 4\sqrt{1-{\sin }^2\theta }\cdot 4\cos \theta d\theta$

$={\int }_0^{\pi /2}\cdot 4\cos \theta \cdot 4\cos \theta d\theta$

$=8{\int }_0^{\pi /2}\cdot 2{\cos }^2\theta d\theta$

$=8{\int }_0^{\pi /2}(1+\cos 2\theta )d\theta$

$=8{\int }_0^{\pi /2}d\theta +8{\int }_0^{\pi /2}\cos 2\theta d\theta$

$=8[\theta {]}_0^{\pi /2}+8\times {[\frac{1}{2}\sin 2\theta ]}_0^{\pi /2}$

$=8\times \frac{\pi }{2}+8\times \frac{1}{2}[\sin 2\times \frac{\pi }{2}-\sin 0^{\circ }]$

$=4\pi +4[\sin \pi -\sin 0^{\circ }]$

$=4\pi +4\times 0=4\pi \text{ Ans: }$