 Hi and welcome to the session. Let us discuss the following question. Question says, choose the correct answer. Integral e raised to the power x multiplied by 1 plus x upon cos square e raised to the power x multiplied by x dx equals A minus cot e multiplied by x raised to the power x plus C B tan x multiplied by e raised to the power x plus C C tan e raised to the power x plus C D cot e raised to the power x plus C Let us now start with the solution. We have to find the integral e raised to the power x multiplied by 1 plus x dx upon cos square of e raised to the power x multiplied by x Now we will solve this integral by substitution method. Clearly we can see if we put e raised to the power x multiplied by x equal to t Then derivative of this function with respect to x is equal to e raised to the power x multiplied by 1 plus x. So we will substitute for a function whose derivative also occurs in the integrand. So we will write put e raised to the power x multiplied by x equal to t Now differentiating both the sides with respect to x we get e raised to the power x multiplied by 1 plus x multiplied by e raised to the power x is equal to dt upon dx Here we have applied product rule to find the derivative of this function Now this can be further written as e raised to the power x multiplied by 1 plus x is equal to dt upon dx Now this implies e raised to the power x multiplied by 1 plus x dx is equal to dt Now we will substitute for the numerator dt in the integrand and in the denominator we will write cos square t We know this function is equal to t We know 1 upon cos t is equal to sec t So 1 upon cos square t is equal to sec t So we can write this integral as integral of sec t dt Now by using the formula of integration we get integral of sec t dt is equal to tan t plus c where c is the constant of integration Now we know t is equal to e raised to the power x multiplied by x So we will substitute e raised to the power x multiplied by x for t in this expression and we get tan of e raised to the power x multiplied by x plus c So we get integral of e raised to the power x multiplied by 1 plus x dx upon cos square of e raised to the power x multiplied by x is equal to tan of e raised to the power x multiplied by x plus c So correct answer is b This is our required answer This completes the session Hope you understood the solution Take care and have a nice day