*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.

*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.

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

Indefinite integral step by step solution sample