 Hello and welcome to the session. Let us discuss the following question. Question says, choose the correct answer. The value of definite integral from 0 to pi upon 2 log of 4 plus 3 sin x upon 4 plus 3 cos x dx is a2b3 upon 4 c0d minus 2. We have to choose the correct answer from a, b, c and d. First of all, let us understand that definite integral from 0 to a fx dx is equal to definite integral from 0 to a f a minus x dx. We will use this property of definite integral as our key idea to solve the given question. Let us now start with the solution. Now we have to find the value of definite integral from 0 to pi upon 2 log of 4 plus 3 sin x upon 4 plus 3 cos x dx. Now let us assume that this integral is equal to i. Now, using that property given in key idea, we can write this integral as definite integral from 0 to pi upon 2 log of 4 plus 3 sin pi upon 2 minus x upon 4 plus 3 cos pi upon 2. Pi upon 2 minus x dx. Now we know sin pi upon 2 minus x is equal to cos x and cos pi upon 2 minus x is equal to sin x. So we get i is equal to log of 4 plus 3 cos x upon 4 plus 3 sin x dx. Now let us name this expression as 1 and this expression as 2. Now adding 1 and 2, we get 2i is equal to definite integral from 0 to pi upon 2 log of 4 plus 3 sin x upon 4 plus 3 cos x dx. Plus definite integral from 0 to pi upon 2 log of 4 plus 3 cos x upon 4 plus 3 sin x dx. Now limits of both the integrals are same. So we can write 2i is equal to definite integral from 0 to pi upon 2 log of 4 plus 3 sin x upon 4 plus 3 cos x plus log of 4 plus 3 cos x upon 4 plus 3 sin x dx. Now we know log m plus log n is equal to log m n. So we can write 2i is equal to definite integral from 0 to pi upon 2 log of 4 plus 3 sin x upon 4 plus 3 cos x multiplied by 4 plus 3 cos x upon 4 plus 3 sin x dx. Now 4 plus 3 cos x will get cancelled by 4 plus 3 cos x and 4 plus 3 sin x will get cancelled by 4 plus 3 sin x. And we get 2i is equal to definite integral from 0 to pi upon 2 log 1 dx. Now we know log 1 is equal to 0. So we get 2i is equal to 0 or we can say i is equal to 0. Now we know i is equal to definite integral from 0 to pi upon 2 log of 4 plus 3 sin x upon 4 plus 3 cos x dx. So we get this integral is equal to 0. So the correct answer is c. So this is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.