Integration of Sine and Cos

Can someone please explain if the book is correct or and I correct? Or why the book does this?

I understand how to integrate Sine and Cos, (e.g: The integral of Sin(5x) would be -0.2Cos(5x)), I understand that. But I'm currently doing Dynamics and all the exmaples in the book do it differently.

Instead, it is slightly different to the usual method. (This is part of an example from a book. Note: V is the velocity in m/s)

Hence the velocity vector is
v(t) = V cos θ + (V sin θ − gt)j
r = ∫ v dt
= ∫ [V cos θ + (V sin θ − gt) ] dt
= Vt cos θ + (Vt sin θ − 0.5gt^2)j + c

What I don't understand is, why do they get Vt cos θ when they integrate V cos θ??? Shouldn't they get Vt Sin θ instead? I checked the answer book and it does this for all of the questions, so it's a bit confusing...


Any help will be greatly appreciated!!!

Notice that in v(t) you have t in parentheses. This means that v is a function of t and we are integrating with respect to t (also notice the 'dt' at the end of the integral).

Here, θ is a constant, and represents the initial angle of trajectory.
V cosθ and V sinθ therefore are just the initial horizontal and vertical components of initial velocity, and are also constant.

So if you wish to be less confused, replace V cosθ with V₁ and V sinθ with V₂, where V₁ and V₂ are constants. Then velocity vector is

v(t) = V₁ i + (V₂ − gt) j

r = ∫ v dt
r = ∫ [V₁ i + (V₂ − gt) j] dt
r = V₁t i + (V₂t − gt²/2) j + c
r = Vt cosθ i + (Vt sinθ − 0.5gt²) j + c

Notice that
r = ∫ vdt.
So you integrate v with respect to t
not θ. Everything that doesn't
have a t in it is a constant.
So ∫ v cos θ dt = v cos θ ∫ dt =
vt cos θ. The same idea works for the
other part of the integral.