integrate sinxcosx please
how does sin (x) cos (x) = (1/2) sin 2x


integration chain rule please help
Started by Stewartd7, May 18 2009 01:29 PM
4 replies to this topic
#1
Posted 18 May 2009 - 01:29 PM
#2
Posted 18 May 2009 - 02:07 PM
because the expansion of sin(2x) is 2sin(x)cos(x)
sin(a+b)=sin(a)cos(b)+cos(a)sin(b), and if a=x,b=x
sin(x+x)=sin(x)cos(x)+cos(x)sin(x)
sin(2x)=2sin(x)cos(x)
sin(x)cos(x)=(1/2)sin(2x)
so the integral is (remember divide by the derivative of the inside
)
-(1/4)cos(2x) +c
sin(a+b)=sin(a)cos(b)+cos(a)sin(b), and if a=x,b=x
sin(x+x)=sin(x)cos(x)+cos(x)sin(x)
sin(2x)=2sin(x)cos(x)
sin(x)cos(x)=(1/2)sin(2x)
so the integral is (remember divide by the derivative of the inside

-(1/4)cos(2x) +c
=-=-=Marcus=-=-=
#3
Posted 18 May 2009 - 02:11 PM
QUOTE (Marcus @ May 18 2009, 03:07 PM) <{POST_SNAPBACK}>
because the expansion of sin(2x) is 2sin(x)cos(x)
sin(a+b)=sin(a)cos(b)+cos(a)sin(b), and if a=x,b=x
sin(x+x)=sin(x)cos(x)+cos(x)sin(x)
sin(2x)=2sin(x)cos(x)
sin(x)cos(x)=(1/2)sin(2x)
so the integral is
-(1/4)cos(2x) +c
sin(a+b)=sin(a)cos(b)+cos(a)sin(b), and if a=x,b=x
sin(x+x)=sin(x)cos(x)+cos(x)sin(x)
sin(2x)=2sin(x)cos(x)
sin(x)cos(x)=(1/2)sin(2x)
so the integral is
-(1/4)cos(2x) +c
where did the half come from???
and how do you do cos^2 (x) then? ta
#4
Posted 18 May 2009 - 02:19 PM
the half comes from dividing both sides by 2, as sin(2x) =2cos(x)sin(x) and we only want one cos(x)sin(x)
and for cos²(x) you use a similar principle only using the cos(2x) expansion
cos(a+b)=cos(a)cos(b)-sin(a)sin(b)
cos(2x)=cos²x-sin², but we know cos²x+sin²x=1 so sin²=1-cos²x and subbing this in
cos(2x)=2cos²x-1
so 2cos²x=cos(2x)+1
cos²=(1/2)(cos(2x)) + (1/2)
which if we integrate, (1/2)(cos(2x)) becomes (1/4)sin(2x) and 1/2 becomes x/2
so final answer
(1/2)((1/2)sin(2x)+x)+c
symbolic answer
and for cos²(x) you use a similar principle only using the cos(2x) expansion
cos(a+b)=cos(a)cos(b)-sin(a)sin(b)
cos(2x)=cos²x-sin², but we know cos²x+sin²x=1 so sin²=1-cos²x and subbing this in
cos(2x)=2cos²x-1
so 2cos²x=cos(2x)+1
cos²=(1/2)(cos(2x)) + (1/2)
which if we integrate, (1/2)(cos(2x)) becomes (1/4)sin(2x) and 1/2 becomes x/2
so final answer
(1/2)((1/2)sin(2x)+x)+c
symbolic answer
=-=-=Marcus=-=-=
#5
Posted 18 May 2009 - 03:06 PM
QUOTE (Marcus @ May 18 2009, 03:19 PM) <{POST_SNAPBACK}>
the half comes from dividing both sides by 2, as sin(2x) =2cos(x)sin(x) and we only want one cos(x)sin(x)
and for cos²(x) you use a similar principle only using the cos(2x) expansion
cos(a+b)=cos(a)cos(b)-sin(a)sin(b)
cos(2x)=cos²x-sin², but we know cos²x+sin²x=1 so sin²=1-cos²x and subbing this in
cos(2x)=2cos²x-1
so 2cos²x=cos(2x)+1
cos²=(1/2)(cos(2x)) + (1/2)
which if we integrate, (1/2)(cos(2x)) becomes (1/4)sin(2x) and 1/2 becomes x/2
so final answer
(1/2)((1/2)sin(2x)+x)+c
symbolic answer
and for cos²(x) you use a similar principle only using the cos(2x) expansion
cos(a+b)=cos(a)cos(b)-sin(a)sin(b)
cos(2x)=cos²x-sin², but we know cos²x+sin²x=1 so sin²=1-cos²x and subbing this in
cos(2x)=2cos²x-1
so 2cos²x=cos(2x)+1
cos²=(1/2)(cos(2x)) + (1/2)
which if we integrate, (1/2)(cos(2x)) becomes (1/4)sin(2x) and 1/2 becomes x/2
so final answer
(1/2)((1/2)sin(2x)+x)+c
symbolic answer
i got 1/4sin(2x+1) +c n is that wrong???
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