 Hi and welcome to the session. I am Purva and I will help you with the following question. Choose the correct answer. Integral cos 2x upon sin x plus cos x whole square dx is equal to a minus 1 upon sin x plus cos x plus c b log sin x plus cos x plus c c log sin x minus cos x plus c and b 1 upon sin x plus cos x whole square. Now we begin with the solution. We denote this integral by i. So we have i is equal to integral cos 2x upon sin x plus cos x whole square dx. Now we put sin x plus cos x whole square equal to dt differentiating this we get 2 into sin x plus cos x into cos x minus sin x dx is equal to dt or we can write this as 2 into cos square x minus sin square x dx is equal to dt because we know that a minus b into a plus b is equal to a square minus b square since a square minus b square is equal to a minus b into a plus b or we can further write this as 2 into now we know that cos square x minus sin square x is equal to cos 2x. So we get cos 2x dx is equal to dt or we can write this as cos 2x dx is equal to 1 by 2 dt. Putting these values in i we get i is equal to 1 by 2 integral dt by t and this further equal to 1 by 2 now integral 1 upon t is log t so we have 1 by 2 log t plus c. Now we know that t is equal to sin x plus cos x whole square so putting the value of t we get this is equal to 1 by 2 log sin x plus cos x whole square plus c or we can write this as this is equal to log sin x plus cos x whole square to the power 1 by 2 plus c. Now cancelling out 2 we get this is equal to log mod sin x plus cos x plus c and this is same as the option b. So we get our answer as b hope you have understood the solution take care and God bless you.