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integration

when i integrate cos(2x)cos(5x)

i obtain the following answer:

1/14 sin(7x) - 1/6 sin(-3x) + c



is this the same as the answer in the textbook>>>

1/14 sin(7x) + 1/6 sin(3x) + c
Reply 1
Avatar for Pheylan
Pheylan
sin(−x)≡−sin(x)sin(-x)\equiv-sin(x)

so yes
Reply 2
Avatar for jayshah31
jayshah31
14
FoOtYdUdE
when i integrate cos(2x)cos(5x)

i obtain the following answer:

1/14 sin(7x) - 1/6 sin(-3x) + c



is this the same as the answer in the textbook>>>

1/14 sin(7x) + 1/6 sin(3x) + c

It is, but for what reason? What is sin⁡(−x)\sin (-x) equal to, i.e. is it an odd or even function?
Reply 3
Avatar for FoOtYdUdE
FoOtYdUdE
OP
9
Pheylan
sin(−x)≡−sin(x)sin(-x)\equiv-sin(x)

so yes

:biggrin: so my original answer's not wrong? :biggrin:
Reply 4
Avatar for TheNack
TheNack
FoOtYdUdE
:biggrin: so my original answer's not wrong? :biggrin:


Your answer was correct, but do you understand why? It is because sin(x) is an odd function.
Reply 5
Avatar for FoOtYdUdE
FoOtYdUdE
OP
9
TheNack
Your answer was correct, but do you understand why? It is because sin(x) is an odd function.

sinx is an odd function? huh? cuz it contains an odd number??? so an even function = sin(4x) etc?? sin(99x) = odd function?
FoOtYdUdE
sinx is an odd function? huh? cuz it contains an odd number??? so an even function = sin(4x) etc?? sin(99x) = odd function?


A function f is even if f(x) = f(-x) and odd if -f(x) = f(-x)

So f(x) = x^2 is even but f(x) = sin(x) is odd.
Reply 7
Avatar for FoOtYdUdE
FoOtYdUdE
OP
9
TheUnbeliever
A function f is even if f(x) = f(-x) and odd if -f(x) = f(-x)

have you made a typo?
FoOtYdUdE
have you made a typo?


No?
Reply 9
Avatar for FoOtYdUdE
FoOtYdUdE
OP
9
TheUnbeliever
No?

erm. nvm i dint undastand wt u wrote tho. thnx anyway.
FoOtYdUdE
erm. nvm i dint undastand wt u wrote tho. thnx anyway.


He's trying to make sure you understand WHY they are same, as you seem happy just to accept they are.

The reason is that sin(x) is an odd function, where an odd function is defined as f(-x) = -f(x). So sin(-x) = -sin(x) as it is odd.

In your case, - 1/6 sin(-3x) is equal to 1/6sin[-(-3x)] which is 1/6 sin(3x).

You need to understand why they're the same for similar questions.
If I were you, I would ignore the odd/even terminology. You simply need to remember that sin(-x) is identical to -sin(x). If you take a look at a graph of y=sin(x) it should be pretty straightforward to see why, but don't get worked up over the reasoning. As long as you know that it is true, you'll be fine.

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