 hello friends let us integrate this function we have to integrate the function sine 4x into sine d 8x with respect to dx now on multiplying and dividing it by 2 we have 1 by 2 integral of 2 sine 4x sine 8x into dx now we know that 2 sine a sine b is equal to cos a minus b minus cos a plus b right on using this formula we have 1 by 2 cos a minus b will be minus 4x minus cos e plus b will be 12x into dx further it can be written as 1 by 2 now cos of minus theta is cos theta so we have cos 4x into dx minus 1 by 2 integral of cos 12x into dx now integral of cos theta is equal to what it is equal to sine theta divided by the derivative of 4x that is 4 minus 1 by 2 into sine 12x divided by 12 plus c which can be taking out 1 by 2 common we are left with 1 by 4 sine 4x minus 1 by 12 sine 12x plus c and this is the required answer to the question that was given to us this completes my session hope you understood it well have a nice day