5.2. THE SECOND FUNDAMENTAL THEOREM OF CALCULUS
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Figure 5.12: Axes for plotting f and F.
(f) Sketch an accurate graph of y = F(x) on the righthand axes provided, and
clearly label the vertical axes with appropriate scale.
⊳
Differentiating an Integral Function
We have seen that the Second FTC enables us to construct an antiderivative F of any
continuous function f by defining F by the corresponding integral function F(x) =
x
c
f (t) dt. Said differently, if we have a function of the form F(x) =
x
c
f (t) dt, then
we know that F ′ (x) =
d
dx
x
c
f (t) dt
= f (x). This shows that integral functions, while
perhaps having the most complicated formulas of any functions we have encountered, are
nonetheless particularly simple to differentiate. For instance, if
F(x) =
x
π
sin(t
2 ) dt,
then by the Second FTC, we know immediately that
F
′ (x) = sin(x
2 ).
Stating this result more generally for an arbitrary function f , we know by the Second
FTC that
d
dx
x
a
f (t) dt
= f (x).
In words, the last equation essentially says that “the derivative of the integral function
whose integrand is f , is f .” In this sense, we see that if we first integrate the function f
from t = a to t = x, and then differentiate with respect to x, these two processes “undo”