 Hi and how are you all today? The question says integrate the following. Here the function is given to us as sin x into 2x into 3x. Here we need to integrate this function with respect to dx. Now here on dividing and multiplying this function by 2 we can have 1 by 2 integral of 2 sin x sin 2x multiplied by sin 3x into dx. Now we know 2 sin a sin b is equal to cos a minus b minus cos a plus b right. So on using this formula we have 1 by 2 integral of here we have a as x and b as 2x so we have cos a minus b that means x minus 2x giving us minus x minus cos a plus b that means 3x into sin 3x into dx and simply find it further we have 1 by 2 cos of minus x is cos x itself so we have cos x minus cos 3x into sin 3x into dx now on opening the brackets we have 1 by 2 cos x into sin 3x minus cos 3x into sin 3x into dx further this can be written as 1 by 2 cos x into sin 3x dx minus 1 by 2 cos 3x into sin 3x to dx further on multiplying and dividing it by 2 we have 1 by 4 to cos x sin 3x dx minus 1 by 4 to cos 3x sin 3x dx. Now here on using the formula that is 2 cos a sin b is equal to sin a plus b that will give us 4x minus sin a minus b that will give us minus 2x right to dx minus 1 by 4 now cos a sin again by using the same formula we have sin a plus b that is 6x minus sin a minus b that will be 0 so here we are left with 1 by 4 integral of sin 6x into dx now further we have 1 by 4 integral of sin 4x dx minus integral of sin minus 2x is minus sin 2x dx sorry 1 by 4 is also there minus 1 by 4 integral of sin 6x into dx now we know that integral of sin x is equal to minus cos x divided by differentiation of x so we have 1 by 4 taking common minus cos 4x divided by 4 here the integral of sin sin 2x will be minus cos so we have minus minus converting into plus and then outside the integral we again have minus so we have cos 2x upon 2 plus cos 6x upon 6 plus c that can be further written as 1 by 4 now on rearranging we have cos 6x upon 6 minus cos 4x upon 4 minus cos 2x upon 2 plus c and the required answer to the question is 1 by 4 into 1 by 6 cos 6x minus 1 by 4 cos 4x minus 1 by 2 cos 2x bracket close plus c right so this completes the session hope you understood it well and have a very nice day ahead