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

1 Answer
Oct 28, 2016

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