 Hello and how are you all today? My name is Priyanka and I shall be helping you with the following question. It says integrate the following. Now here the fourth function which is given to us is sin cube 2x plus 1 where 2x plus 1 is the theta. Now here we need to integrate the function sin cube 2x plus 1 with respect to dx. Now here we should know the formula for sin cube theta first and it is equal to 1 by 4 3 sin theta minus sin 3 theta. Now here we have theta equal to 2x plus 1 right. So on using the formula we have integral of 1 by 4 3 sin theta that is 2x plus 1 minus sin 3 theta that is 2x plus 1 into dx. Taking out 1 by 4 out of the integral we have 1 by 4 integral of 3 sin 2x plus 1 minus sin 3 2x plus 1 into dx. Further we can write it as now here the integral of sin theta is cos theta into derivative sorry divided by the derivative of theta that is 2x plus 1 derivation. So we have 1 by 4 into 3. Integral of sin theta is cos theta divided by the differentiation of theta that is 2 minus 1 by 4 here integral of sin 3 theta will be equal to here we have minus sin also minus cos 3 theta divided by the differentiation of 3 theta that will be 3 2x plus 1 that is 3 into 2 plus c. Now on simplifying it we have minus 3 by 8 cos 2x plus 1 plus cos 3 2x plus 1 divided by 24 plus c. Further we know that cos 3 theta can be written as 4 cos cube theta minus 3 cos theta. So here we have minus 3 by 8 cos 2x plus 1 plus 1 by 24 now here cos 3 2x plus 1 can be written as 4 cos cube theta that is 2x plus 1 minus 3 cos theta that is 2x plus 1 plus c. Now here we can write because cos 3 theta is equal to 4 cos cube theta minus 3 cos theta. Further on further simplifying we have minus 3 by 8 cos 2x plus 1 plus this can be opened the bracket 1 by 6 cos cube 2x plus 1 minus 1 by 8 cos 2x plus 1 plus c. Now here after simplifying we have minus 3 by 8 cos 2x plus 1 minus 1 by 8 cos 2x plus 1 can be written as minus 1 by 2 cos 2x plus 1 plus 1 by 6 cos cube 2x plus 1 plus c and this is the required answer to the solution to the question. Right this completes the session understood it well have a nice day.