 Hi and how are you all today? The question says prove that integral 0 to fx dx is equal to integral 0 to a f of a minus x dx. Hence evaluate integral 0 to pi by 2 dx upon 1 plus tan x. So here in this question first we need to prove and then we need to evaluate. First of all let us start with our right hand side of this. Let us put a minus x equal to t. So we have dx is equal to minus dt right? And also when x is equal to 0 then t will be equal to a and when x is equal to a then t will be equal to 0 right? So we have integral 0 to a f of a minus x dx written as a to 0 f of t into minus dt. This further gives us minus a to 0 f of t dt. And this is by property 1 we can write this as integral 0 to a f of x dx which is same as our left hand side right? So left hand side is equal to right hand side and hence we have proved first part for this question. Now we need to evaluate this value. Now here when tan x is written as sin x upon cos x it becomes 0 to pi by 2 cos x upon sin x plus cos x dx. Let this be equal to i. Then i is equal to 0 to pi by 2 we can write it as cos pi by 2 minus x upon cos pi by 2 minus x plus sin pi by 2 minus x dx. Right we know that cos pi by 2 minus x is sin x upon cos pi by 2 minus x is again sin x plus cos x dx. Let this be the first equation this be the second equation. Now on add the first and the second equation we get i plus i is equal to integral 0 to pi by 2 cos x upon sin x plus cos x dx plus 0 to pi by 2 sin x upon sin x plus cos x. This implies 2y is equal to 0 to pi by 2 cos x plus sin x that is what combining these two upon sin x plus cos x dx. This further implies 2i is equal to 0 to pi by 2 1 dx this 2y equal to x these are the limits that we are going to use that is 2y is equal to pi by 2 minus 0 that gives us i is equal to pi by 4. Right so the value of 0 to pi by 2 dx upon 1 plus tan x is equal to pi by 4. Right so this completes our solution hope you understood it well i enjoyed it too have a nice day ahead.