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integration chain rule please help - HSN forum

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integration chain rule please help


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#1 Stewartd7

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Posted 18 May 2009 - 01:29 PM

integrate sinxcosx please

how does sin (x) cos (x) = (1/2) sin 2x

#2 Marcus

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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 biggrin.gif)
-(1/4)cos(2x) +c
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#3 Stewartd7

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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




where did the half come from???

and how do you do cos^2 (x) then? ta

#4 Marcus

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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
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#5 Stewartd7

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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



i got 1/4sin(2x+1) +c n is that wrong???






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