hello friends!so we are going to integrate sin^3x so there is an odd power of sinx so we have to convert into cos to make it easy! so lets take sinx and sin^2x and split them out so it will give me back sin^2x times sinxdx so you can see that i have just split them out and nothing else so i have to find some way to convert sin^2x into cos^2x and thats not difficult ,its simple! so i have to just write 1-cos^2x instead of sin^2x and sinx is same as above now you may be thinking how we get this it is just a trigonometric identity which you may have studied in your 10th std sin^2x +cos2^x=1 we may prove this but right now we are not going to jump into it so let us see so how i am going to solve this so it is 1-cos^2x so if we take u=cosx we would get du=-sinxdx which will make it simple! so lets take u=cosx.Therefore,du=-sinxdx so my sinxdx is right there(in above step). Therefore,-du=sinxdx.which we have right there(in above step) so it will give me (1-cos^2x).. let us write it cos^2x right now. times (-du) so now we have u^2=cos^2x. so lets plug everything into the integral so we get (1-u^2)times(-du) and now it is very simple integral so just split the minus inside the terms so it is (u^2-1)timesdu so it will give me u^3/3-u+c now u is nothing but cosx right there so it is same as cos^3x/3-cosx+c do not forget to write (+c) because it is an indefinite integral so it is the answer..!!!