Question

Given the function: f(x) = cos 2x
We get the primitive function F(x) sin 2x / 2

Why divided by two? It seems to me the answer should just be sin 2x.

Answer 1

If someone could explain the rule / rules this follow, that would be much appreciated. All I know is that the primitive function of cos x is sin x +C and the derivatine of sin x is cos x. I don't see where the division by two comes from...

Answer 2

To find the primitive of cos 2x, you think of what function has cos 2x as a derivative.
Easy: that is sin 2x, isn't it? Almost!
If you check: (sin 2x)' = cos 2x * 2. This is because of the Chain Rule: the function sin 2x consists of two separate functions. According to the Chain Rule, you differentiate sin 2x. This becomes cos 2x. Then you multiply with the derivative of 2x, which is 2.

All in all: (sin 2x)'= 2cos2x.

So: sin 2x + C is NOT the primitive of cos 2x, but ½sin2x +C is: because of the ½, this will compensate for the extra factor 2 that you don't need...

Answer 3

Hmm, I'll take some time to digest this. Thank you!

Answer 4

Try this: what is a primitive of sin102x ?
I mostly do a rough guess first: it MUST be something like cos102x + C.

The I realize, (cosx)'= -sinx, so I change my answer to -cos102x + C.

Now check it: \((-\cos102x + C)'= --\sin102x \cdot 102=102 \sin102x\).

Hmm...that's not what I want, it is 102 times too big! Therefore change it to:

\(-\frac{1}{102}\cos102x+C\). Now I know everything will work out fine!

Answer 5

Allright, I've got it. Help much appreciated :)

Answer 6

YW!