The role of constant of integration

Question

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?

Answer 1

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?

Answer 2

One interesting thing is to take a look at the expansion of $e^x$: $$e^x=\sum_{k\ge0}\frac{x^k}{k!}$$If we take the integral of both sides, we get $$\sum_{k\ge1}\frac{x^k}{k!}+C$$Notice the change in the index. This is because the number $1$ becomes $x$ after integration. But because of the constant of integration, $e^x$ is the antiderivative of itself.