int_{pi / 4}^{pi / 3} frac{cos x-sin x}{sin 2 | vedclass

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Question

(A) माना $I = \int_{\pi / 4}^{\pi / 3} \frac{\cos x-\sin x}{\sin 2 x} d x$. हम जानते हैं कि $\int \frac{\cos x-\sin x}{\sin 2 x} dx = \int \frac{\cos

Vedclass · Accepted Answer

$\frac{1}{2} \log \left[\frac{(3+2 \sqrt{2})(2-\sqrt{3})}{\sqrt{3}}\right]$

Vedclass · Answer

(A) माना $I = \int_{\pi / 4}^{\pi / 3} \frac{\cos x-\sin x}{\sin 2 x} d x$. हम जानते हैं कि $\int \frac{\cos x-\sin x}{\sin 2 x} dx = \int \frac{\cos x}{2 \sin x \cos x} dx - \int \frac{\sin x}{2 \sin x \cos x} dx = \frac{1}{2} \int \csc x dx - \frac{1}{2} \int \sec x dx$. अतः,$I = \frac{1}{2} [\log |	an(x/2)| - \log |\sec x + 	an x|]_{\pi/4}^{\pi/3} = \frac{1}{2} [\log |\frac{	an(x/2)}{\sec x + 	an x}|]_{\pi/4}^{\pi/3}$. $x = \pi/3$ पर मान $\frac{2-\sqrt{3}}{\sqrt{3}}$ प्राप्त होता है और $x = \pi/4$ पर मान $3-2\sqrt{2}$ प्राप्त होता है। इस प्रकार,$I = \frac{1}{2} \log \left( \frac{(2-\sqrt{3})/\sqrt{3}}{3-2\sqrt{2}} ight) = \frac{1}{2} \log \left( \frac{(2-\sqrt{3})(3+2\sqrt{2})}{\sqrt{3}} ight)$,जो विकल्प $A$ है।

$\int_{\pi / 4}^{\pi / 3} \frac{\cos x-\sin x}{\sin 2 x} d x=$

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