Calculus II - Trigonometric Integrals

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Homework Statement



Apparently I'm doing something wrong. I'm kind of lost as to what because I looked over my work several times.

Homework Equations



sin^2 x = ( 1 - cos 2x )/2
cos^2 x = ( 1 - sin 2x )/2
integral sin(x) dx = -cos(x)
integral cos(x) dx = sin(x)

The Attempt at a Solution



SEE ATTACHMENT
Wolfram Alpha says the correct solution is 1/64 (24 x+8 sin(4 x)+sin(8 x))+constant
Thanks for any help you can provide!
 

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  • #2
eumyang
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sin^2 x = ( 1 - cos 2x )/2
cos^2 x = ( 1 - sin 2x )/2
The bolded is not right. It should be
[tex]\cos^2 x = \frac{1 + \cos 2x}{2}[/tex]
 
  • #3
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Thanks for your response. My book specifically told me those were true and even use the fact that they were to evaluate a integral. For example:
 
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  • #5
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So i guess there's a plus sign and I forgot to include it becuase i used minus instead

sin^2 x = ( 1 - cos 2x )/2
cos^2 x = ( 1 + sin 2x )/2

cos^2 x = ( 1 + sin 2x )/2 instead of cos^2 x = ( 1 - sin 2x )/2

That is one mistake in my work... assuming that formula is right
 
  • #6
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oh man im an idiot thanks
 
  • #7
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I guess I'm still doing something wrong, don't know what though
 

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  • #8
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cos^2(x) = 1/2(1 + cos(2x))
 
  • #9
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Ya I know. I still don't see what I did wrong though in post #7
 
  • #10
eumyang
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Ya I know. I still don't see what I did wrong though in post #7
It's actually the same as the answer you got from Wolfram. You just need to factor out another 1/8.
[itex]\frac{1}{8}\left( 3x + \sin 4x + \frac{\sin 8x}{8}\right) + C[/itex]
[itex]= \frac{1}{8}\left( \frac{24x}{8} + \frac{8\sin 4x}{8} + \frac{\sin 8x}{8}\right) + C[/itex]
Do you see it now?
 
  • #11
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The answers don't seem to match though for some reason and I don't have a clue as to what I'm doing wrong...

The answer I got was
1/8 [ 3x + sin(4x) + sin(8x)/8 ]+ c
wolfram alpha returned
1/32 (12 x-8 sin(2 x)+sin(4 x))+constant

if they are equal to each other I should be able to plug in a value for x in both expressions and get the same results... this however doesn't occur

1/8 [ 3*5 + sin(4*5) + sin(8*5)/8 ]+ c is about 2.000760549 + c

1/32 (12*5 -8 sin(2*5)+sin(4*5))+constant is about 2.039534817 + c

It would appear to be that i am off by a little bit... which is strange cause I don't see what I did wrong

and ya they would appear to be equal to each other but i guess they are not
 
  • #13
Ray Vickson
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The answers don't seem to match though for some reason and I don't have a clue as to what I'm doing wrong...

The answer I got was
1/8 [ 3x + sin(4x) + sin(8x)/8 ]+ c
wolfram alpha returned
1/32 (12 x-8 sin(2 x)+sin(4 x))+constant

if they are equal to each other I should be able to plug in a value for x in both expressions and get the same results... this however doesn't occur

1/8 [ 3*5 + sin(4*5) + sin(8*5)/8 ]+ c is about 2.000760549 + c

1/32 (12*5 -8 sin(2*5)+sin(4*5))+constant is about 2.039534817 + c

It would appear to be that i am off by a little bit... which is strange cause I don't see what I did wrong

and ya they would appear to be equal to each other but i guess they are not
For *indefinite* integrations, the C is an arbitrary constant. There is no reason the two C's should be the same. However, you ought to get the same results for the definite integral from a to b. Have you tried it?

RGV
 
  • #14
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Ya I got different results when I integrated both functions from 2 to 10

for 1/8 [ 3*x + sin(4*x) + sin(8*x)/8 ]
I got roughly 18.01464011

for 1/32 (12*x -8 sin(2*x)+sin(4*x))
I got roughly 18.13678944
 
  • #15
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The answers don't seem to match though for some reason and I don't have a clue as to what I'm doing wrong...

The answer I got was
1/8 [ 3x + sin(4x) + sin(8x)/8 ]+ c
wolfram alpha returned
1/32 (12 x-8 sin(2 x)+sin(4 x))+constant
WA gave me 1/64(24x + 8sin(4x) + sin(8x)) + constant, which is equivalent to the answer you say you got above.
For the other answer that you say WA gave you, I took its derivative in WA and it gave sin4x, so it looks like you had a mistake when you entered it in.
 
  • #16
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huh weird thanks
 

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