I would like to know the whole purpose of adding a constant termed constant of integration everytime we integrate an indefinite integral $\int f(x)dx$. I am aware that this constant "goes away" when evaluating definite integral $\int_{a}^{b}f(x)dx $. What has that constant have to do with anything? Why is it termed as the constant of integration? Where does it come from?
The motivation for asking this question actually comes from solving a differential equation $$x \frac{dy}{dx} = 5x^3 + 4$$ By separation of $dy$ and $dx$ and integrating both sides, $$\int dy = \int\left(5x^2 + \frac{4}{x}\right)dx$$ yields $$y = \frac{5x^3}{3} + 4 \ln(x) + C .$$
I've understood that $\int dy$ represents adding infinitesimal quantity of $dy$'s yielding $y$ but I'am doubtful about the arbitrary constant $C$.
Answer
Sometimes you need to know all antiderivatives of a function, rather than just one antiderivative. For example, suppose you're solving the differential equation
$$
y'=-4y
$$
and there's an initial condition $y(0)=5$. You get
$$
\frac{dy}{dx}=-4y
$$
$$
\frac{dy}{y} = -4\;dx
$$
$$
\int\frac{dy}{y} = \int -4\;dx
$$
$$
\log_e y = -4x + C.
$$
Here you've added a constant. Then
$$
y = e^{-4x+C} =e^{-4x}e^C
$$
$$
5=y(0)= e^{-4\cdot0}e^C = e^C
$$
So
$$
y=5e^{-4x}.
$$
(And sometimes you need only one antiderivative, not all of them. For example, in integration by parts, you may have $dv=\cos x\;dx$, and conclude that $v=\sin x$.)
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