what is the antiderivative of -1/5 sin5xdx?

Question

i just need to know how to integrate when sin, cos etc have a value before the x......

yeeppp i got that part.... but why did the answer get to 1/25?

Answer 1

I = (-1/5) ∫ sin (5x) dx
Let u = 5x
du = 5 dx
dx = du / 5
I = (-1/25) ∫ sin u du
I = (1/25) cos u + C
I = (1/25) cos 5x + C

Answer 2

The answer in this case is 1/25 cos5x (+c).

I think the simplest way to understand it is to use the chain rule:

d/dx cos(u) = -sin(u) du/dx

in this case u = 5x and du/dx = 5.

Thus d/dx cos5x = -5sin5x