Question

How do you find the antiderivative of `cos(5x)`?

Answer 1

Say that:

`y=sin(kx)` whereby k is a constant.

Now, transform this into:

`y=sin(u)` whereby `u=kx`.

If this is the case:

`(dy)/(du)=cos(u)=cos(kx)`

`(du)/(dx)=k`

And this means that:

`(dy)/(du)*(du)/(dx)=(dy)/(dx)=kcos(kx)`

Now, when k=5:

When `y=sin(5x)`, `(dy)/(dx)=5cos(5x)`.

Ok... So what happens when:

`y=1/5sin(5x)`??

Well...

`5y=sin(5x)`

Differentiating both sides of the equation and remembering that `(dy)/(dy)*(dy)/(dx)=(dy)/(dx)` you get...

`5*(dy)/(dx)=5cos(5x)`

Then dividing both sides of this equation by 5 you get...

`(dy)/(dx)=cos(5x)`

This means that:

`intcos(5x)dx=1/5sin(5x)+C`