Question

Question #e1803

Answer 1

`intsin3xdx=-1/3cos3x+C`

Explanation:

by the chain rule

`d/(dx)(cos3x)=-3sin3x`

and as integration is the reverse of differentiation

`intsin3xdx=-1/3cos3x+C`

Answer 2

`-(1/3)cos3x + C`

Explanation:

Since we are dealing with integration, we must always include a "+C" as the derivative of any Constant is 0

Questions to ask:

The derivative of what is `sinx`?

Well, if you know your trig rules, we know that the derivative of `cosx` is `-sinx`, so we can conclude that the derivative of `-cosx` is `sinx`.

What about the `3x`?

Well, since there is a #`3x` inside the cos , we can infer that the chain rule is involved:

Ex.

`d/dx sin4x = (cos4x)(4)`

`d/dx cos6x = -(sin6x)(6)`

So, knowing the chain rule, we know that when this was derived, the `3x` was derived and multiplied by the entire equation, So we must ask:

If the `3` is multiplied by the equation, then why is there a `1` in front instead of a `3`?

For this to be true there must have been a number already in front of the `-cos3x` so:

`3(?) = 1`

`3(1/3) = 1`

So now we know that, originally, there was a (1/3) in front of the equation, so we conclude that:

`intsin(3x)dx = -(1/3)cos3x + C`