 Hello and how are you all today? The question says integrate the following. Now the function which is given to us is cos 2x cos 4x cos 6x all getting multiplied by each other. Now before proceeding on with the solution we should be well versed with the formula which says 2 cos a cos b can be written as cos e plus b plus cos a minus b and the knowledge of this formula is the key idea towards this question. Proceeding on, now here we need to find out the integral of cos 2x cos 4x let us keep these two in bracket getting multiplied by cos 6x. Now here on applying the above mentioned formula we have multiplying and dividing this function by 2. We have 1 by 2 out of the integral of 2 cos 2x cos 4x in one bracket multiplied by cos 6x dx that is further equal to 1 by 2. Now here cos 2x into cos 4x will become cos e and b where a is 2x and b is 4x. So we have on applying the formula as cos e plus b that means 2x plus 4x that is 6x plus cos 2x that is 2x minus 4x and we also know that cos of minus theta is cos theta. So we have plus 2 cos x getting multiplied by cos 6x dx. Now here this can be written as 1 by 2 integral of cos 6x will get multiplied by both the terms so we have cos square 6x dx plus 1 by 2 cos 2x further 1 by 2 integral of cos square 6x dx can be written as 1 plus cos 2 theta where theta is 6x divided by 2 dx plus 1 by it will become 4 to have a 2 getting multiplied by the numerator function cos 2x cos 6x dx. Further we have the answer as 1 by 2 integral of 1 by 2 plus cos 12x by 2 dx plus 1 by 4 2 cos a cos b can be written as cos a plus b plus cos a minus b so it will be minus 4x that will be a positive 4x as cos of minus x is cos x only. Further we have 1 by 2 now this function can be written as 1 by 2 1 by 2 dx plus 1 by 2 integral of cos 12x by 2 dx plus 1 by 4 cos 8x dx plus 1 by 4 cos 4x dx. Further on simplifying we have the answer as 1 by 4 integral of dx plus 1 by 4 integral of cos 12x dx plus 1 by 4 integral of cos 8x dx plus 1 by 4 integral of cos 4x dx. Now finding out the integral of all of them we have 1 by 4 integral of dx is x plus 1 by 4 integral of cos 12x dx will be equal to what it will be sin 12x divided by the derivative of 12x that is 12 plus 1 by 4 into again the integral of cos 8x will be equal to sin 8x divided by 8 plus 1 by 4 into integral of sin 4x divided by the derivative of 4x that is 4. Now taking 1 by 4 common we have the final answer 1 by 4 into sin 12x plus here we have 1 by 8 sin 8x plus 1 by 4 is taken out common so we are left with 1 1 by 4 so plus 1 by 4 sin 4x plus c right so this is the required answer to the session hope you enjoyed and understood the concept have a nice day.