Indefinite integral, basic definitions:

Indefinite integral is the operation, inverse to the differentiation. So, the task of indefinite integration defined very simple: given the function f(x), find function F(x) such as:

F'(x) = f(x) (1)

Note, that relation (1) will not change, if one adds to the function F(x) some arbitrary constant:

(F(x) + Const)' = F'(x) + (Const)' = F'(x) + 0 = f(x),

because its derivative equals to zero. Hence, indefinite integral is defined to arbitrary constant.

Definition:

Indefinite integral of the given function f(x) is called set of all its antiderivatives.

∫ f(x) dx=F(x) +Const (2)

In the equation (2):
∫ - indefinite integral symbol,
f(x) - integrand (subintegral function),
dx - differential,
F(x) - antiderivative (for function f(x)),
Const - arbitrary constant.

Primitives table:

To get primitives table one need to read the derivatives table from right to left. For instance, it is known that:

(sin(x))'= cos(x)

then:

∫ cos(x) dx=sin(x)+Const