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# ∫ sinx cosx dx .. two results?!! watch

1. I am trying to do ∫ sinx cosx dx
which I can do, but the confusing thing is I can get two results!

if I use substitution
u = sin x
du = cos x dx

then ∫ u du

which is (u^2)/2
i.e. (sin^2 x)/2

but.. what if I used u=cos x?
du = -sin x dx

and -∫ u du
i.e. (-cos^2) /2

which one is right and why? It can't be both can it? Have I done something wrong?
2. (Original post by kimoni)
I am trying to do ∫ sinx cosx dx
which I can do, but the confusing thing is I can get two results!

if I use substitution
u = sin x
du = cos x dx

then ∫ u du

which is (u^2)/2
i.e. (sin^2 x)/2

but.. what if I used u=cos x?
du = -sin x dx

and -∫ u du
i.e. (-cos^2) /2

which one is right and why? It can't be both can it? Have I done something wrong?
Neither...Do you remember what sin2x is equal to??????2sinxcos
1/2 sin2x
3. You may just have different C (the constant) values.
4. oh yeah..
but why can't I do it my way? shouldnt that work too?
I checked on calc101.com and it said my second answer is right.. what confuses me is why the first one isnt
5. (Original post by kimoni)
I am trying to do ∫ sinx cosx dx
which I can do, but the confusing thing is I can get two results!

if I use substitution
u = sin x
du = cos x dx

then ∫ u du

which is (u^2)/2
i.e. (sin^2 x)/2

but.. what if I used u=cos x?
du = -sin x dx

and -∫ u du
i.e. (-cos^2) /2

which one is right and why? It can't be both can it? Have I done something wrong?
(sin^2x)/2 + A =1/2 -1/2(cos^2x) +A =-(cos^2x/2)+B.
so they differ by a constant.
6. Integrate them both between limits. You get the same answers. ie they are both right.
7. (Original post by SUKBarracuda)
Integrate them both between limits. You get the same answers. ie they are both right.
That's not mathematically too cool.
8. (Original post by fishpaste)
That's not mathematically too cool.
in some cases it is (but not in this case)
9. (Original post by SUKBarracuda)
Integrate them both between limits. You get the same answers. ie they are both right.
ah ok, thats good to know

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