 Hello and welcome to the session. I am Shashi and I am going to help you with the following question. Question says, find the following integrals. We have to find integral of 2x square plus e raised to the power x dx. First of all let us understand that integral of fx plus dx dx is equal to integral of fx dx plus integral of gx dx and integral of k fx dx where k is any real number is equal to k multiplied by integral of fx dx also integral of x raised to the power n dx is equal to x raised to the power n plus 1 upon n plus 1 plus c where c is any constant and integral of e raised to the power ex dx is equal to e raised to the power x plus c where c is any constant. Now these four expressions are the key idea to solve the given question. Let us now start with the solution. We have to find the integral of 2x square plus e raised to the power x dx. Now using expression one of the key idea we can write this integral as integral of 2x square dx plus integral of e raised to the power x dx. Now clearly we can see 2 is a real number. So we can write this integral as 2 multiplied by integral of x square dx plus integral of e raised to the power x dx. Here we have used second expression of the given key idea. Now we know integral of x square is equal to x raised to the power 2 plus 1 upon 2 plus 1. Here we have already shown that integral of x raised to the power n dx is equal to x raised to the power n plus 1 upon n plus 1 plus c where c is any constant. Now here value of n is 2. So we get integral of x square is equal to x raised to the power 2 plus 1 upon 2 plus 1. And here we have written that integral of e raised to the power x dx is equal to e raised to the power x plus c. So integral of this function is equal to e raised to the power x and c is the constant of integration. Now this is further equal to 2 upon 3 multiplied by x cube plus e raised to the power x plus c. So required integral of the given function is equal to 2 upon 3 x cube plus e raised to the power x plus c. This is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.