 Hi and welcome to the session. I am Purva and I will help you with the following question. If f of a plus b minus x is equal to f of x, then integral limit from a to b, x into f of x dx is equal to a, a plus b by 2 integral limit from a to b, f of b minus x dx, b, a plus b by 2 integral limit from a to b, f of b plus x dx, c, b minus a by 2 integral limit from a to b, f of x dx, d, a plus b by 2 integral limit from a to b, f of x dx. Let us now begin with the solution. Let us denote this definite integral by i. So we have i is equal to integral limit from a to b, x into f of x dx. Now let us assume that a plus b minus x is equal to z. So let a plus b minus x is equal to z. Differentiating we get minus dx is equal to dz. When x is equal to a that is lower limit we have z is equal to b because putting here a in place of x we get z is equal to b and when x is equal to b that is upper limit we have z is equal to a because putting here b in place of x we get z is equal to a. Putting all these values in i we get i is equal to minus integral limit from b to a, a plus b minus z into f of a plus b minus z dz. This is equal to integral limit from a to b, a plus b minus z into f of z dz. Now we know that f of a plus b minus x is equal to f of x given to us. So f of a plus b minus z is equal to f of z since f of a plus b minus x is equal to f of x. Now we can write this as this is equal to integral limit from a to b, a plus b into f of z dz minus integral limit from a to b, z into f of z dz. Now this is further equal to integral limit from a to b, a plus b into f of x dx minus integral limit from a to b x into f of x dx right. This is equal to integral limit from a to b, a plus b into f of x dx minus i because we have denoted i is equal to integral limit from a to b x into f of x dx. We can write this as this implies i plus i is equal to a plus b integral limit from a to b f of x dx. This implies 2i is equal to a plus b integral limit from a to b f of x dx and this further implies that i is equal to a plus b by 2 integral limit from a to b f of x dx. Therefore we get i is equal to a plus b by 2 integral limit from a to b f of x dx and this is equal to the option d. Therefore we get our answer as d. Hope you have understood the solution. Take care and bye.