 Hello and welcome to the session. Let us discuss the following question. Question says, choose the correct answer integral 10 multiplied by x raised to the power 9 plus 10 raised to the power x multiplied by log 10 to the base e upon x raised to the power 10 plus 10 raised to the power x dx equals 10 raised to the power x minus x raised to the power 10 plus c, b 10 raised to the power x plus x raised to the power 10 plus c, c 10 raised to the power x minus x raised to the power 10, 10 whole raise to the power minus 1 plus c d log 10 raise to the power x plus x raise to the power 10 plus c. Let us now start with the solution. Now we have to find integral of 10 multiplied by x raise to the power 9 plus 10 raise to the power x log 10 to the base e upon x raise to the power 10 plus 10 raise to the power x dx. Clearly we can see this numerator is the derivative of denominator here. Now we will integrate this function by substitution method of integration. We know we make a substitution for a function whose derivative also occurs in the integrand. So we will make substitution for this function that is x raise to the power 10 plus 10 raise to the power x. So we can write put x raise to the power 10 plus 10 raise to the power x is equal to t. Now differentiating both the sides with respect to x we get 10 multiplied by x raise to the power 9 plus 10 raise to the power x log 10 to the base e is equal to dt upon dx. We know derivative of this term is equal to 10 x raise to the power 9 and let us understand how we will get derivative of this term. Let us assume that y is equal to 10 raise to the power x. Now taking log on both the sides we get log y is equal to log of 10 raise to the power x. Now this can be further written as x log 10. So we get log y is equal to x log 10. Now differentiating both the sides with respect to x we get 1 upon y dy upon dx is equal to x multiplied by 1 upon 10 multiplied by 0 plus log 10 to the base e multiplied by 1. Now this term will become 0 and we get 1 upon y multiplied by dy upon dx is equal to log 10 to the base e. Now this implies dy upon dx is equal to y multiplied by log 10 to the base e. Now substituting value of y here that is 10 raise to the power x we get dy upon dx is equal to 10 raise to the power x log 10 to the base e. So we get derivative of 10 raise to the power x is equal to 10 raise to the power x log 10 to the base e. Now this further implies 10 multiplied by x raise to the power 9 plus 10 raise to the power x multiplied by log 10 to the base e dx is equal to dt. Now substituting these values in this integrand we get dt upon t we know x raise to the power 10 plus 10 raise to the power x is equal to t and this complete function present in the numerator is equal to dt. So we can write it as integral of dt upon t. Now this is further equal to log t plus c. Using the formula of integration that integral of dx upon x is equal to log x plus c we get integral of dt upon t is equal to log t plus c where c is the constant of integration. Now we know t is equal to x raise to the power 10 plus 10 raise to the power x. So here we can write x raise to the power 10 plus 10 raise to the power x plus c. So integral of this given function is equal to log of x raise to the power 10 plus 10 raise to the power x plus c where c is the constant of integration. So our correct answer is d. So this is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.