The constant of integration

Why every indefinite integral carries a "+C", what that constant actually represents, and how extra information (a point on the curve, an initial condition) pins it down.

Pure · Integration

The constant of integration

Why every indefinite integral carries a "+C", what that constant actually represents, and how extra information (a point on the curve, an initial condition) pins it down.

Why it works

Differentiation destroys information. The functions

x^2

x^2 + 5

and

x^2 - 100

all have exactly the same derivative,

2x

— differentiating wipes out any constant term, because a constant changes nothing about the slope of a graph.

Integration is the reverse of differentiation. So when you ask "what function has derivative

2x

?", the honest answer is: a whole family of functions —

x^2 + C

for any constant

C

. You cannot know which member of the family is "the" answer from the derivative alone, because the derivative never contained that information in the first place.

Picture it: the graphs of

x^2 + C

for different

C

are infinitely many copies of the same parabola, stacked vertically. At any given

x

, every one of them has the same gradient. The derivative tells you the shape; the

+C

records which copy you're on.

That's why the

+C

is not decoration or a mark-scheme ritual — it is the honest admission that one piece of information is missing. And it's why one extra fact (the curve passes through

(1, 4)

; the particle starts at rest;

v = 3

when

t = 0

) is exactly enough to find

C

: it picks out the one curve from the family that fits.

In a definite integral the

+C

vanishes because you compute

[F(x) + C]_a^b = (F(b) + C) - (F(a) + C)

— the two

C

s cancel. You are measuring a difference between two points on the same curve, and every curve in the family gives the same difference.