 Hi and welcome to the session, let us discuss the following question, question says, show that definite integral from 0 to a fx gx dx is equal to 2 multiplied by definite integral from 0 to a fx dx if f and g are defined as fx is equal to fa minus x and gx plus ga minus x is equal to 4. Let us now start with the solution. Now we are given that fx is equal to fa minus x and gx plus ga minus x is equal to 4. Now this implies gx is equal to 4 minus ga minus x. Now definite integral from 0 to a fx gx dx is equal to definite integral from 0 to a f a minus x 4 minus ga minus x dx. Clearly we can see fx is equal to fa minus x. So we can write fx as fa minus x and we know gx is equal to 4 minus ga minus x. So we can substitute 4 minus ga minus x for gx. Now this further implies definite integral from 0 to a fx gx dx is equal to definite integral from 0 to a f a minus x dx multiplied by 4 minus definite integral from 0 to a f a minus x multiplied by g a minus x dx. We can write this integral as difference of these two integrals. Now we know this integral can be written as 4 multiplied by definite integral 0 to a fx dx. Here we have used this property of definite integrals and we will write this integral as it is that is definite integral from 0 to a f a minus x g a minus x dx. Now let us consider integral from 0 to a f a minus x ga minus x dx. Now put a minus x equal to y. Now differentiating both the sides with respect to x we get minus dx is equal to dy. The new limits are when x is equal to 0, y is equal to a and when x is equal to a y is equal to 0. Now we can write this integral is equal to definite integral from a to 0 f y g y minus dy. We know a minus x is equal to y and dx is equal to minus dy. And as x varies from 0 to a y varies from a to 0. Now this integral is further equal to negative of definite integral a to 0 f y g y dy. Now it is further equal to definite integral from 0 to a f y g y dy. Here we have used this property of definite integral. Now using this property of definite integral this integral is further equal to definite integral from 0 to a f x g x dx. So we get definite integral from 0 to a f a minus x ga minus x dx is equal to definite integral from 0 to a f x dx dx. Now let us name this expression as 1. Now we will substitute this integral for this integral in expression 1. Now expression 1 becomes definite integral from 0 to a f x dx dx is equal to 4 multiplied by definite integral from 0 to a f x dx minus definite integral from 0 to a f x dx dx. Now this implies 2 multiplied by definite integral from 0 to a f x dx dx is equal to 4 multiplied by definite integral from 0 to a f x dx. Adding this integral on both the sides of this expression we get this expression. Now dividing both the sides by 2 we get definite integral from 0 to a f x dx dx is equal to 2 multiplied by definite integral from 0 to a f x dx. Now this completes our proof. So we can write hence proved this completes the session. Hope you understood the solution. Take care and have a nice day.