Integration may be thought as the inverse problem of differentiation: instead of differentiating a function

we're looking for a function whose derivative is the function given:

where $k$ is some constant of choice, say $0$, $456$, $-\dfrac{3}{7}$, anything. And it works!

Wow! We say that

is the indefinite integral of

But how on earth could we compute this humongous function? Well, that's a crucial difference between differentiation and integration: to differentiate, one has to follow a set of rules, whereas to integrate, there is no set of rules that work for every function. One can only have a set of tricks that may work, and we'll study these rules through the chapter

Why the $k$? The derivative of a function is always unique, but this is not the case with integration:

Instead it is unique up to a constant, because if two functions have the same derivative, its difference has to be a constant

Once a candidate for integral is found,

adding a constant will preserve this quality

so we always include this constant $k$ symbolically

and thus the integral is called indefinite. And why all this funny stuff of $\displaystyle\int$ and $\mathrm{d}x$? We'll talk about it when we study definite integrals in the next chapter.