MATHEMATICS  THEORY



This section introduces the Fundamental Theorem of Calculus. 





Fundamental Theorem of Calculus



The concept of
derivative and
integral will
be related by the Fundamental Theorem of Calculus. This theorem consists of two parts.
 Part one states:
If function f is continuous on interval between a and b, then the function g which is defined by
a ≤ x ≤ b
is continuous on the [a,b], differentiable on (a,
b), and g^{ '}(x) = f(x).
 Part two states:
If function f is continuous on the interval between a and b, then
(1)
where F is any antiderivative of f, that is F^{ '}(x) = f(x).
The above equation can also be expressed as




Continuous Function f(t)
= t^{2} + t 

This theorem can be better understood by finding the
result of .
The function t^{2} + t is a continuous function over the interval
between 3 and x. According to the first part of Fundamental Theorem of
Calculus, the derivative of this function is






In order to double check the answer, can
be determined by an alternative method.
First, use the second part of Fundamental Theorem of Calculus to find
the antiderivative of on the interval [3, x].




Function 
Antiderivative 
∫cf(x)dx 
cF(x) + c 
∫x^{n}dx (n ≠ 1) 
x^{n+1}/(n + 1) + c 
∫(f(x) + g(x))dx 
F^{}(x) + G^{}(x) + c 
∫cosxdx 
sinx + c 
∫sinxdx 
cosx + c 
∫sec^{2}xdx 
tanx + c 
∫secxtanxdx 
secx + c 
∫cscxcotxdx 
cscx + c 
∫csc^{2}xdx 
cotx + c 


Next, take the derivative of the result gives
It is clear that both methods give the same results.
In order to use the Fundamental Theorem of Calculus
effectively,
important
formulas
of antiderivative are given in the table on the left. 


