 Hello and welcome to the session. In this session, we discussed the following question which says evaluate integral of e to the power x into sine 4x minus 4 whole upon 1 minus cos 4x t whole dx. Before we move on to the solution, let's discuss some results. First we have sine 2x is equal to 2 sine x into cos x, then cos 2x is equal to 1 minus 2 sine square x. Also, integral of e to the power x into f of x plus f dash x the whole dx is equal to e to the power x into fx plus c where the c is the constant of integration. This is the key idea that we use in this question. Let's proceed with the solution now. We are supposed to evaluate the integral which is supposed to be equal to i. So, let i be equal to the integral e to the power x into 4x minus 4 upon 1 minus cos 4x the whole dx. So, first of all we will expand sine 4x and cos 4x using this formula. This gives us i is equal to integral e to the power x into now sine 4x can be written as 2 sine 2x into cos 2x minus 4 upon 1 minus cos 4x can be written as 1 minus 2 sine square this whole dx. So, further we get i is equal to integral of e to the power x into cos 2x minus 4 upon when we open these parenthesis we are left with 2 sine square 2x the whole dx. Now, further we can write i is equal to integral e to the power x into 2x into cos 2x upon i square 2x minus 4 upon i square 2x the whole dx. Now, here 2 cancels with 2 sine 2x cancels with sine 2x and so we have i is equal to integral of e to the power x into cos 2x upon sine 2x minus now here sine square 2x the whole. We now have i is equal to integral e to the power x into cos 2x upon sine 2x would be cos 2x minus 2 into cos x square 2x the whole dx. So, as we know 1 upon sine x is now d by dx is equal to minus 2 which means that the derivative of cos 2x is this term which is minus 2 cos x square 2x. We suppose f of x to be equal to cos 2x so this means that f dash x is equal to minus 2 cos x square to integral of e to the f of x plus the whole dx. So, from the key idea that integral of e to the power x into fx plus f dash x dx is equal to e to the power x fx plus c thus this is equal to e to the power x into fx plus c. And therefore we can say that i is equal to e to the power x into fx which is cos 2x where again this c is the constant of integration thus integral of e to the power x into minus cos 4x the whole dx is equal to e to the power x into cos 2x plus this is our final answer. This completes the session. Hope you have understood the solution of this question.