ক. $\int \sin x^\circ dx = \int \sin (\frac{\pi x}{180}) dx$। $$= -\frac{180}{\pi} \cos (\frac{\pi x}{180}) + c = -\frac{180}{\pi} \cos x^\circ + c$$ খ. $f(x) = 4-x^2$। ধরি $x = 2\sin\theta$, $dx = 2\cos\theta d\theta$। $$\begin{aligned} \int_{-1}^1 x^2 \sqrt{4-x^2} dx &= \int_{-\pi/6}^{\pi/6} (2\sin\theta)^2 (2\cos\theta) (2\cos\theta) d\theta \\ &= 16 \int_{-\pi/6}^{\pi/6} \sin^2\theta \cos^2\theta d\theta = 2 \int_{-\pi/6}^{\pi/6} (1-\cos 4\theta) d\theta \\ &= 2 [\theta - \frac{\sin 4\theta}{4}]_{-\pi/6}^{\pi/6} = \frac{2\pi}{3} - \frac{\sqrt{3}}{2} \end{aligned}$$ গ. বৃত্ত $x^2+y^2=25$ এবং রেখা $x=3$। ক্ষেত্রফল $= 2 \int_3^5 \sqrt{25-x^2} dx$। ধরি $x=5\sin\theta$, $dx=5\cos\theta d\theta$। $$\begin{aligned} A &= 2 \int_{\sin^{-1}(3/5)}^{\pi/2} 5\cos\theta \cdot 5\cos\theta d\theta = 25 \int_{\sin^{-1}(3/5)}^{\pi/2} (1+\cos 2\theta) d\theta \\ &= 25 [\theta + \frac{\sin 2\theta}{2}]_{\sin^{-1}(3/5)}^{\pi/2} = \frac{25\pi}{2} - 25\sin^{-1}(3/5) - 12 \end{aligned}$$