 Hi and welcome to the session, I am Purva and I will help you with the following question. Evaluate the definite integral, integral limit from pi by 2 to pi e raised to the power x into 1 minus sin x upon 1 minus cos x dx. Now let us begin with the solution now. Let us denote this definite integral by i. So we have i is equal to integral limit from pi by 2 to pi e raised to the power x into 1 minus sin x upon 1 minus cos x dx. This is equal to integral limit from pi by 2 to pi e raised to the power x into 1 minus. Now we know that sin 2 theta is equal to 2 sin theta cos theta. So we can write sin x as 2 sin theta x by 2 into cos x by 2 upon 1 minus. Now cos 2 theta is equal to 1 minus 2 sin square theta. So we can write cos x as 1 minus 2 sin square x by 2. Write dx since sin 2 theta is equal to 2 sin theta cos theta and cos 2 theta is equal to 1 minus 2 sin square theta. This is equal to integral limit from pi by 2 to pi e raised to the power x into 1 minus 2 sin x by 2 into cos x by 2 upon 1 minus 1. Now here minus into minus will become plus. So we get plus 2 sin square x by 2 dx. Now here in the denominator plus 1 and minus 1 will cancel out. So we get this is equal to integral limit from pi by 2 to pi e raised to the power x into 1 upon 2 sin square x by 2 dx minus integral limit from pi by 2 to pi e raised to the power x into 2 sin x by 2 into cos x by 2 upon 2 sin square x by 2 dx. Write now we can write this as this is equal to 1 upon 2 integral limit from pi by 2 to pi e raised to the power x into now 1 upon sin square x by 2 can be written as cos x square x by 2. So we get cos x square x by 2 dx minus integral limit from pi by 2 to pi e raised to the power x into now here 2 and 2 will cancel out in numerator and denominator and sin x by 2 and sin x by 2 will cancel out in numerator and denominator. So we get cos x by 2 upon sin x by 2 which is equal to cot x by 2. So we get cot x by 2 dx. Now here taking e raised to the power x as the first function and cos x square x by 2 as the second function and applying Bipart's method we get this is equal to 1 upon 2 into first function that is e raised to the power x into integration of second that is integration of cos x square x by 2 which is equal to minus cot x by 2 upon 1 by 2 minus integration derivative of first that is e raised to the power x into integration of second that is minus cot x by 2 upon 1 by 2 and limit is from pi by 2 to pi dx minus integral limit from pi by 2 to pi e raised to the power x into cot x by 2 dx. Now opening the bracket we get this is equal to 1 by 2 into now 1 upon 1 by 2 will give us 2. So we get 2 into e raised to the power x into minus cot x by 2 limit is from pi by 2 to pi minus 1 by 2 into 2 integral limit from pi by 2 to pi e raised to the power x into minus cot x by 2 dx minus integral limit from pi by 2 to pi e raised to the power x into cot x by 2 dx. This is equal to now cancelling out 2 in numerator and denominator we get minus e raised to the power x into cot x by 2 limit is from pi by 2 to pi minus again cancelling out 2 in numerator and denominator and here minus into minus will become plus so we get plus integral limit from pi by 2 to pi e raised to the power x into cot x by 2 dx minus integral limit from pi by 2 to pi e raised to the power x into cot x by 2 dx. This is equal to now putting the limits we get minus e raised to the power pi into cot pi by 2 minus minus of e raised to the power pi by 2 into cot pi by 4 because here we have upper limit as pi so putting the upper limit that is pi in place of x we get minus e raised to the power pi into cot pi by 2 minus minus of putting lower limit that is pi by 2 in place of x we get e raised to the power pi by 2 into cot pi by 4. Now these two integrals will cancel out because of positive and negative sign so we get this is equal to zero because we know that cot pi by 2 is equal to zero so we get here zero and here minus into minus will become plus so we get plus e raised to the power pi by 2 into now cot pi by 4 is equal to 1 so we get a 1 so we get this is equal to e raised to the power pi by 2 thus we get our answer as e raised to the power pi by 2 hope you have understood the solution buy and take care