THE FUNDAMental THEOREM OF CALCULUS ILLUSTRATIONS OF THE FUNDAMENTAL THEOREM In this section, we will illustrate both parts of the Fundamental Theorem by applying them in solving various integrals. Examples 1 through 12 will deal with Part 1 of the Theorem, while examples 13 to 19 will deal with Part 2. To be able to evaluate an integral using FTC 2, you need to know how to compute an antiderivative for any given function. The table below lists some general functions and their corresponding antiderivative formulas:
Function
General Antiderivative
f(x) + g(x)
F(x) + G(x)
cf(x)
cF(x)
cos x
sin x
sec 2 x
tan x
sec x tan x
sec x
x (n ≠ -1)
xn+1 n+1
n
Study the following examples carefully.
Example 1 Using Part 1 of the Fundamental Theorem, compute the derivative of
g(x)
=
∫
x
√ 1 + 2t
0
dt
Solution
Always remember this: to differentiate an integral, the first step should be to ensure that the integrand is a continuous function; because a discontinuous function IS NOT differentiable. Thus, since the integrand f(t) = 1 + 2t is continuous, the FTC 1 gives:
g '(x)
=
√ 1 + 2x
Example 2
Using Part 1 of the Fundamental Theorem, compute the derivative of
g(x)
=
∫ (2 + t ) x
1
4
5
dt
Solution The integrand f(t) = (2 + t4)5 is continuous. Therefore, using FTC 1 gives its derivative as:
g'(x)
=
(2 + x4)5