int(costheta+sintheta)/(costheta-sintheta)d theta এর মান কত? 

প্রশ্ন: \(\int \frac{\cos\theta + \sin\theta}{\cos\theta - \sin\theta} d\theta\) এর মান কত?

সমাধান:

ধরি, \(I = \int \frac{\cos\theta + \sin\theta}{\cos\theta - \sin\theta} d\theta\)

এখানে, হর \(\cos\theta - \sin\theta\)-কে \(z\) ধরলে, লব \(\cos\theta + \sin\theta\) পাওয়া যায়।

ধরি, \(z = \cos\theta - \sin\theta\)
তাহলে, \(\frac{dz}{d\theta} = -\sin\theta - \cos\theta = -(\sin\theta + \cos\theta)\)
সুতরাং, \(dz = -(\sin\theta + \cos\theta) d\theta\)
অতএব, \((\sin\theta + \cos\theta) d\theta = -dz\)

এখন, \(I = \int \frac{-dz}{z} = -\int \frac{1}{z} dz = -\ln|z| + c\)
\( = -\ln|\cos\theta - \sin\theta| + c\)

আমরা জানি, \(\cos\theta - \sin\theta = \sqrt{2} \left(\frac{1}{\sqrt{2}}\cos\theta - \frac{1}{\sqrt{2}}\sin\theta\right)\)
\( = \sqrt{2} \left(\cos\frac{\pi}{4}\cos\theta - \sin\frac{\pi}{4}\sin\theta\right)\)
\( = \sqrt{2} \cos\left(\theta + \frac{\pi}{4}\right)\)

সুতরাং, \(I = -\ln\left|\sqrt{2} \cos\left(\theta + \frac{\pi}{4}\right)\right| + c\)
\( = -\ln\left|\sqrt{2}\right| - \ln\left|\cos\left(\theta + \frac{\pi}{4}\right)\right| + c\)
\( = -\frac{1}{2}\ln 2 - \ln\left|\cos\left(\theta + \frac{\pi}{4}\right)\right| + c\)
\( = - \ln\left|\cos\left(\theta + \frac{\pi}{4}\right)\right| + c_1\), যেখানে \(c_1 = c -\frac{1}{2}\ln 2\)

আমরা জানি, \(-\ln|\cos x| = \ln|\sec x|\)
সুতরাং, \(I = \ln\left|\sec\left(\theta + \frac{\pi}{4}\right)\right| + c_1\)

অতএব, \(\int \frac{\cos\theta + \sin\theta}{\cos\theta - \sin\theta} d\theta = \ln\left|\sec\left(\theta + \frac{\pi}{4}\right)\right| + c_1\)

উত্তর: \(\log_e \sec(\theta + \frac{\pi}{4}) + c\) 🎉