Hi i no from formula sheets that the integral of cosec^2x equals -cotx but i still cant work out how to prove this integral.
so far iv tried a few different things
1. i no that cosec^2=1/sin^2x but this doesnt help much
2. cosec^2x=cot^2x + 1, again... couldn't get much further with that
any help would be greatly appreciated thanks
Hey, I thought I'd help you out as I've just done this problem myself. Use translation: as sin(x)=cos((pi/2)-x) cosec(x)=sec((pi/2)-x) so cosec^2(x)=sec^2((pi/2)-x) this is easily integratable as I(sec^2(x))=tanx: I(sec^2((pi/2)-x))=-tan(pi/2-x)+k = -cot(x)+k
Hey, I thought I'd help you out as I've just done this problem myself. Use translation: as sin(x)=cos((pi/2)-x) cosec(x)=sec((pi/2)-x) so cosec^2(x)=sec^2((pi/2)-x) this is easily integratable as I(sec^2(x))=tanx: I(sec^2((pi/2)-x))=-tan(pi/2-x)+k = -cot(x)+k
I doubt that dude will ever come back on here again lol.
BUT I might as well ask you to note that his question asked how to get the integral of cosec2(x) without having prior knowledge that -cot(x) differentiates to it. In your explanation, you use a change of trig functions which is fine but the one part that you haven't explained is how ∫sec2(x)dx=tan(x). Of course, we know that if we differentiate tan(x), we get sec^2(x) but this does not solve his original question.
If you want to look at how you could, have a look at t-substitutions.