 Hello and welcome to the session. My name is Mansi and I am going to help you with the following question. The question says integrate the following function that is e raised to part 2x into sin x. So let us see the solution to this question. We have to find integral i that is equal to integral of e raised to part 2x sin x dx. Now according to the i-layer rule we see that trigonometric function is given preference over exponential function for becoming the first function. So sin x will act as the first function and e raised to part 2x as the second function. So now by integration by parts we have first function that is sin x into integral of second function that is e raised to part 2x dx minus integral of d by dx of first function that is sin x into integral of e raised to part 2x dx. Now this is equal to sin x into integral of e raised to part 2x will be e raised to part 2x divided by differentiation of 2x that is 2 minus integral of d by dx of sin x is cos x. Again integral of this is same as this e raised to part 2x divided by 2 into dx. Now this will be equal to e raised to part 2x divided by 2 sin x minus integral of, now here we will have just this much minus integral of e raised to part 2x by 2 cos x dx here also we have dx. Now again on applying integration by parts here we will take this as the first function and e raised to part 2x as the second function will take 1 by 2 out of the integral sin because that is the constant. So we have e raised to part 2x divided by 2 sin x minus 1 by 2 into, now this is the first function so we have cos x into e raised to part 2x divided by 2 minus integral of d by dx of first function will be minus sin x into integral of e raised to part 2x dx the whole into dx. This will be equal to e raised to part 2x by 2 sin x minus 1 by 2 into cos x into e raised to part 2x divided by 2 now minus sin comes out of the integral we have plus integral of sin x into e raised to part 2x divided by 2 dx. This is equal to e raised to part 2x by 2 sin x minus 1 by 2 into cos x into e raised to part 2x by 2 again 1 by 2 comes out as a constant and we have integral sin x into e raised to part 2x dx now we see that e raised to part 2x into sin x dx is same as the integral i so this is same as i therefore we can say that i is equal to e raised to part 2x by 2 sin x minus 1 by 4 into cos x into e raised to part 2x minus 1 by 4i now we take this to the left hand side and we get i plus 1 by 4i is equal to e raised to part 2x by 2 sin x minus 1 by 4 cos x into e raised to part 2x plus a constant c now we have to add these two quantities first of all we take the lcm of the denominators that is 4 in the numerator we will have 4i plus i that is 5i is equal to e raised to part 2x by 2 into sin x minus 1 by 4 cos x into e raised to part 2x plus now we call this constant as c1 so here we have plus c1 now on multiplying throughout by 4 we get 5i is equal to e raised to part 2x into sin x into 2 minus cos x into e raised to part 2x plus some other constant c this implies that i is equal to we take e raised to part 2x common divided by 5 is 1 by 5 is common and we will have 2 sin x minus cos x plus c so our answer to this question is e raised to part 2x divided by 5 into 2 sin x minus cos x plus c so i hope that you understood the question and enjoyed the session have a good day