 Hello and welcome to the session I am Deepika here. Let's discuss the question which says integrate the following function e raised to power x into 1 plus sin x over 1 plus cos x. Let us first understand the integral of the time e raised to power x into fx f dash x and this is equal to e raised to power x plus c. This is the key idea behind our question. We will take the help of this key idea to solve the above question. So let's start the solution. Now our given function is e raised to power x into 1 plus sin x over 1 plus cos x and we have to integrate it. Now this integral can be written in the form e raised to power x into 1 over 1 plus cos x plus sin x over 1 plus cos x. We have a formula 1 plus cos 2 theta is equal to 2 cos square theta and sin 2 theta is equal to 2 sin theta cos theta. So our given integral is equal to e raised to power x into 1 over 1 plus cos x. Now 1 plus cos 2 theta is 2 cos square theta so 1 plus cos x is 2 cos square theta by 2 plus. Now sin 2 theta is 2 sin theta cos theta so sin x is 2 sin x by 2 cos x by 2 over again 2 cos square x by 2. Now this is again equal to integral of e raised to power x into 1 over 2. Now 1 over cos theta is sin theta so 1 over cos square x by 2 is sin square x by 2. Now sin theta upon cos theta is tan theta so sin x by 2 upon cos x by 2 is tan x by 2. Again we can rewrite this integral as e raised to power x 1 over 2 sin square x by 2 equal to 1 by 2 the given e raised to power x into 1 plus over 1 plus cos x. The idea integral of e raised to power x into fx plus f dash x is equal to e raised to power x solution is clear to you.