Why do I get different results for the same integral?

Question

The variables $ a, b, s, c $ are constants, so:

$$ \int \left ( a \cos(s + cx) - b \sin(s + cx) \right ) dx = \frac{a\sin(s + cx) + b\cos(s + cx)}{c} +C $$

But if $c=0$ then:

$$ \int \left ( a \cos(s) - b \sin(s) \right ) dx = x( a\cos(s) + b\sin(s) ) +C $$

How is that possible? Why is the result different? At $c=0$ the first equation makes no sense, because the $c$ is in the denominator.

Answer 1

Let the constant of integration be $C+(-a\sin s-b\cos s)/c$. The first answer becomes $$\frac{a(\sin(s+cx)-\sin s)+b(\cos(s+cx)-\cos s)}c+C\\=\frac{2a\cos (s+\frac{cx}2)\sin\frac{cx}2-2b\sin(s+\frac{cx}2)\sin\frac{cx}2}c+C$$ The fundamental trigonometric limit $$\lim_{c\to0}\frac{\sin\frac{cx}2}c=\frac x2$$ lets you simplify my second line to your answer (with a sign error).

Answer 2

That is because when you use $0$ as your constant inside the function, that whole function becomes a constant and is multiplied by $x$ for the integration.