Q:

The role of constant of integration

Let us consider the following integral
$$\int \frac{\sin x}{\cos^3 x}dx\, .$$
If we note that $\tan x= \frac{\sin x}{\cos x}$, we have
$$\int \frac{\tan x}{\cos^2 x}dx = \frac{\tan^2 x}{2}+C \quad (1)$$
since $\int f(x) f^\prime (x) dx= \frac{f^2(x)}{2}+C$ and $f(x)=\tan x$.
Otherwise, if we substitute $t=\cos x$:
$$-\int\frac{dt}{t^3}=\frac{1}{2\cos^2 x}+C \quad (2) .$$
The result shall be the same, but $\frac{\tan^2 x}{2}\neq \frac{1}{2\cos^2 x}$. The reason for this apparent difference is the constant $C$ in (1) and (2). They are not the same constant, and actually $\frac{1}{\cos^2 x}=\tan^2 x+1$. If we denote by $C_1$ the constant of (1) and by $C_2$ the constant of (2), we have $C_2=\frac12+C_1$. In my opinion, this example is useful to let students know how important is the constant of integration.
Can someone suggest me other examples of this type, without trigonometric functions?

A:

You can also refer to these other interesting cases

Where did I go wrong with my odd proof that $\frac{3dx}{3x} = \frac{5dx}{5x} \iff 3=5$?
The constant of integration during integration by parts and With integration by parts, why don't we write the constant of integration when integrating the second function inside the integral?

