 Hello and welcome to the session. In this session we discussed the following question which says evaluate integral 0 to pi e to the power cos x upon e to the power cos x plus e to the power minus cos x dx. Before we move on to the solution let's recall one important property of definite integrals according to which we have integral 0 to a fx dx is equal to integral 0 to a f of a minus x dx. This is the key idea to be used for this question. Let's move on to the solution now. We take let i be equal to the given integral 0 to pi e to the power cos x upon e to the power cos x plus e to the power minus cos x dx. Now using this property stated in the key idea we have i is equal to integral 0 to pi e to the power cos of pi minus x upon e to the power cos of pi minus x plus e to the power minus cos of pi minus x dx. Now first of all let's take this as equation 1. Now next this i would be equal to integral 0 to pi e to the power minus cos x since we know that cos of pi minus x is equal to minus cos x. This upon now e to the power cos of pi minus x again would be e to the power minus cos x plus e to the power minus of cos pi minus x would be e to the power cos x dx. Let this be equation 2. Now further adding equations 1 and 2 we get 2 i is equal to integral 0 to pi e to the power cos x upon e to the power cos x plus e to the power minus cos x dx plus integral 0 to pi e to the power minus cos x upon e to the power minus cos x plus e to the power cos x dx. So this is further equal to integral 0 to pi e to the power cos x plus e to the power minus cos x upon e to the power cos x plus e to the power minus cos x dx. This is 2 i. So further 2 i is equal to integral 0 to pi 1 into dx since this cancels with this. So we get 2 i is equal to x limit goes from 0 to pi. So we have 2 i is equal to pi minus 0 plus 2 i is equal to pi or you can say i is equal to pi pi 2 and i is the given integral therefore the given integral 0 to pi e to the power cos x upon e to the power cos x plus e to the power minus cos x dx is equal to pi by 2. So this is our final answer. This completes the session. Hope you have understood the solution of this question.