গণিত প্রশ্ন

AI-তৈরি নমুনা উত্তর

ধরি,
$I = \int \frac{dx}{\cos{x} + \sin{x}}$

হরকে $\sqrt{1^2 + 1^2} = \sqrt{2}$ দ্বারা গুণ ও ভাগ করে পাই,
$I = \frac{1}{\sqrt{2}} \int \frac{dx}{\frac{1}{\sqrt{2}}\cos{x} + \frac{1}{\sqrt{2}}\sin{x}}$
$= \frac{1}{\sqrt{2}} \int \frac{dx}{\cos{x}\cos{\frac{\pi}{4}} + \sin{x}\sin{\frac{\pi}{4}}}$
$= \frac{1}{\sqrt{2}} \int \frac{dx}{\cos{\left(x - \frac{\pi}{4}\right)}}$
$= \frac{1}{\sqrt{2}} \int \sec{\left(x - \frac{\pi}{4}\right)} dx$
$= \frac{1}{\sqrt{2}} \ln{\left| \sec{\left(x - \frac{\pi}{4}\right)} + \tan{\left(x - \frac{\pi}{4}\right)} \right|} + c$
(যেখানে $c$ একটি সমাকলন ধ্রুবক)

উত্তর: $\frac{1}{\sqrt{2}} \ln{\left| \sec{\left(x - \frac{\pi}{4}\right)} + \tan{\left(x - \frac{\pi}{4}\right)} \right|} + c$ অথবা $\frac{1}{\sqrt{2}} \ln\left| \tan\left(\frac{x}{2} + \frac{\pi}{8}\right) \right| + c$