समाकलन int frac{left(1-frac{1}{sqrt{3}}right) | vedclass

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Question

(A) माना $I = \int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{1+\frac{2}{\sqrt{3}} \sin 2 x} dx = \int \frac{(\sqrt{3}-1)(\cos x-\sin x)}

Vedclass · Accepted Answer

$\frac{1}{2} \log _{ e }\left|\frac{	an \left(\frac{ x }{2}+\frac{\pi}{12}ight)}{	an \left(\frac{ x }{2}+\frac{\pi}{6}ight)}ight|+ C$

Vedclass · Answer

(A) माना $I = \int \frac{\left(1-\frac{1}{\sqrt{3}}ight)(\cos x-\sin x)}{1+\frac{2}{\sqrt{3}} \sin 2 x} dx = \int \frac{(\sqrt{3}-1)(\cos x-\sin x)}{\sqrt{3}+2\sin 2x} dx$. अंश और हर को $\frac{1}{2}$ से गुणा करने पर: $I = \int \frac{(\frac{\sqrt{3}}{2} - \frac{1}{2})(\cos x-\sin x)}{\sin 60^{\circ} + \sin 2x} dx$. $\sin 60^{\circ} + \sin 2x = 2 \sin(x + \frac{\pi}{6}) \cos(x - \frac{\pi}{6})$ का उपयोग करने पर: $I = \int \frac{\cos(x + \frac{\pi}{6}) - \sin(x - \frac{\pi}{6})}{2 \sin(x + \frac{\pi}{6}) \cos(x - \frac{\pi}{6})} dx$. $I = \frac{1}{2} \int \left( \frac{1}{\sin(x + \frac{\pi}{6})} - \frac{1}{\cos(x - \frac{\pi}{6})} ight) dx$. समाकलन करने पर: $I = \frac{1}{2} \ln \left| \frac{	an(\frac{x}{2} + \frac{\pi}{12})}{	an(\frac{x}{2} + \frac{\pi}{6})} ight| + C$.

समाकलन $\int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(1+\frac{2}{\sqrt{3}} \sin 2 x\right)} d x$ का मान ज्ञात कीजिए।

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