 Hi and welcome to the session, let us discuss the following question, question says, if f x is equal to definite integral from 0 to x t sin t dt then f dash x is cos x plus x sin x b x sin x c x cos x d sin x plus x cos x. We have to choose the correct answer from a, b, c and d. First of all let us understand that if we are given any function f x and f dash x is derivative of this function then f x is equal to n t derivative of f dash x. This is the key idea to solve the given question. Let us now start with the solution. Now we are given f x is equal to definite integral from 0 to x t sin t dt. First of all let us consider integral t sin t without limits. So we can write integral t sin t dt is equal to t multiplied by integral of sin t dt minus integral of minus cos t dt. Now in this step we have done integration by using integration by parts method. Let this function is first function and this function is second function and we also know that the integral of the product of two functions is equal to first function multiplied by integral of the second function minus integral of differential coefficient of first function multiplied by integral of second function. We know differential coefficient of first function that is t is equal to 1 and integral of sin t is equal to minus cos t. Now this integral is equal to minus cos t. So here we can write t multiplied by minus cos t plus minus and minus sin will multiply and they will form a positive sign and here we can write integral of cos t dt. Now this is further equal to minus t cos t plus sin t you know integral of cos t dt is equal to sin t. Now definite integral 0 to x t sin t dt is equal to minus t cos t plus sin t limits from 0 to x value of this function at upper limit x is equal to minus x cos x plus sin x and value of this function at lower limit 0 is equal to 0 plus sin 0 we know sin 0 is equal to 0. So this bracket will become 0 only and we get minus x cos x plus sin x. We are given that f x is equal to 0 to x t sin t dt and this integral is equal to minus x cos x plus sin x. So we can write f x is equal to minus x cos x plus sin x we are required to find f dash x. So differentiating both the sides with respect to x we get f dash x is equal to minus 1 multiplied by cos x plus minus x multiplied by minus sin x plus cos x we know derivative of sin x is cos x and here we have applied product rule to find derivative of this function. Now simplifying further we get minus cos x plus x sin x plus cos x is equal to f dash x. Now minus cos x and plus cos x will get cancelled and we get f dash x is equal to x sin x. So the correct answer is B. So B is our required answer. This completes the session. Hope you understood the solution. Take care and keep smiling.