 Hello and welcome to the session. Let us discuss the following question, question says, find the following integrals. We have to find integration of 4 e raised to the power 3 x plus 1 dx. First of all let us understand that integral of function fx plus dx with respect to x is equal to integral of function fx with respect to x plus integral of function gx with respect to x. Also integral of e raised to the power ax dx is equal to e raised to the power ax upon a plus c where c is any constant and integral of dx is equal to x plus c where c is any constant. Now these three expressions give the key idea to solve the given question. Let us now start with the solution. Now we have to find integral of the function 4 e raised to the power 3 x plus 1 with respect to x. Now using expression 1 of key idea we can write this integral as integral of 4 e raised to the power 3 x dx plus integral of dx. Now this integral can be further written as 4 multiplied by integral of e raised to the power 3 x dx plus integral of dx. Here we have used the property that if k is any real number then integral of kfx dx is equal to k multiplied by integral of fx dx. Now from expression 2 of the key idea we get integral of e raised to the power 3 x is equal to e raised to the power 3 x upon 3 plus constant. So we can write here e raised to the power 3 x upon 3 plus c1 and integral of dx is equal to x. So here we will write x plus c2 where c2 and c1 both are constants. Now this expression can be further written as 4 upon 3 e raised to the power 3 x plus x plus c. If we substitute c4 c1 plus c2 we get this expression. So we get integral of 4 e raised to the power 3 x plus 1 is equal to 4 upon 3 e raised to the power 3 x plus x plus c. This is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.