Mr. Smith’s CALC
CALC Assignment 4.2: Constant of Integration
Recall that working with anti-derivatives was simply our way of “working backwards”
In determining anti-derivatives, we were simply looking to find out what equation we
started with in order to produce the derivative that was before us
Ex. Find the anti-derivative of a(t) = 3t - 6e2t
Definitions:
An anti-derivative of f(x) is any function F(x) such that F`(x) = f(x)  If F(x) is any antiderivative of f(x) then the most general anti-derivative of f(x) is called an indefinite
integral and denoted  f(x)dx = F(x) + C where C is any constant
In this definition the  is called the integral symbol, f(x) is called the integrand, x is
called the integration variable and the “C” is called the constant of integration  So
we can interpret the statement  f(x)dx as “determine the integral of f(x) with respect
to x”
The process of finding an indefinite integral (or simply an integral) is called integration.
(x4 + 3x – 9)dx =
(x2x)dx =
(x4 + 3x – 9)dx =
(cos + 2sin3)d =
e2xdx =
(8x + sec2x)dx =
sin(2x)dx =
(2 - x)2dx =
Now for that Pesky Constant of Integration “C”.
Given that f(x)dx = F(x) + C, we can determine a specific function if we knew what
C was equal to  so if we knew something about the function F(x), then we could
solve for C
Evaluate (x3 – 3x + 1)dx if F(0) = -2
F(x) = ¼x4 – 3/2x2 + x + C
Since F(0) = -2 = ¼(0)4 – 3/2(0)2 + (0) + C
So C = -2 and
F(x) = ¼x4 – 3/2x2 + x - 2
Position, Velocity, Acceleration
Mr. Smith’s CALC
—An object moves along a co-ordinate line with a velocity v(t) = 2 - 3t + t2
meters/sec. Its initial position is 2 m to the right of the origin.
—(a) Determine the position of the object 4 seconds later
—(b) Determine the total distance traveled in the first 4 seconds