bIntegration/b is the reverse process of differentiation. The notation for an bindefinite/b integral is:

(\displaystyle\int{f'(x)} dx = f(x) + c )

Indefinite integrals always produce a bfunction/b.

bDefinite/b integrals are covered in topic=8/90Topic 8.3/topic.

[b]uThe constant of integration[/u]/b

Consider the equations below:

(\textcolor{blue}{y=x^3-2x+2} \quad \textcolor{red}{y=x^{3}-2x} \quad \textcolor{green}{y=x^{3}-2x-2} )

mtaimg/images/topics/8/8-85-1.png/mtaimg

The derivatives of all three equations are the same:

(\dfrac{dy}{dx} = 3x^2-2 )

So when performing the reverse process (integration), it is impossible to know which equation was the starting point.

A bconstant of integration/b ((\bm{c})) is required to represent the (y)-intercept. Hence:

(\displaystyle\int{(3x^2-2)} dx = x^3-2x + \bm{c} )

(\implies y=x^3-2x+c)

If you know a point that the original equation goes through, then you can substitute the (x) and (y)-coordinates into the equation and solve for (c).

[b]uIntegration[/u]/b

(\displaystyle\int{f'(x)} dx = f(x) + c )