Improper Integrals
MATH 211, Calculus II
J. Robert Buchanan
Department of Mathematics
Fall 2021
,Definite Integrals
Theorem (Fundamental Theorem of Calculus (Part I))
If f is continuous on [a, b] then
Z b
f (x) dx = [F (x)]x=b
x=a = F (b) − F (a)
a
where F is any antiderivative of f on (a, b).
Z 1
1
Question: Can we evaluate the definite integral dx?
−1 x2
, Answer
We cannot use the Fundamental Theorem of Calculus to
evaluate Z 1
1
2
dx
−1 x
since the integrand has a discontinuity at x = 0. If we try to
evaluate it using the Fundamental Theorem of Calculus we get
1
1 x=1
Z
1
dx = − = −2
−1 x2 x x=−1
a result which is impossible since 1/x 2 > 0 for −1 ≤ x < 0 and
0 < x ≤ 1.
MATH 211, Calculus II
J. Robert Buchanan
Department of Mathematics
Fall 2021
,Definite Integrals
Theorem (Fundamental Theorem of Calculus (Part I))
If f is continuous on [a, b] then
Z b
f (x) dx = [F (x)]x=b
x=a = F (b) − F (a)
a
where F is any antiderivative of f on (a, b).
Z 1
1
Question: Can we evaluate the definite integral dx?
−1 x2
, Answer
We cannot use the Fundamental Theorem of Calculus to
evaluate Z 1
1
2
dx
−1 x
since the integrand has a discontinuity at x = 0. If we try to
evaluate it using the Fundamental Theorem of Calculus we get
1
1 x=1
Z
1
dx = − = −2
−1 x2 x x=−1
a result which is impossible since 1/x 2 > 0 for −1 ≤ x < 0 and
0 < x ≤ 1.
