Introduction
The function F is called an antiderivative of f if F 0 (x) = f (x). Furthermore, any other antiderivative of
f will take the form F (x) + k for some constant k. For this reason, we call
F (x) + C,
where C is an arbitrary constant, the most general
antiderivative of f . The indefinite integral provides
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notation for antiderivatives. The notation “ f (x) dx00 represents the most general antiderivative of f . The
statements
Z
d
F (x) = f (x) and
f (x) = F (x) + C
dx
are equivalent.
Common antiderivatives we need to know include:
Common Antiderivatives
Z
Z
1
1
xn+1 + C, n 6= −1
•
dx = ln |x| + C
• xn dx =
n+1
x
Z
Z
• sin(x) dx = − cos(x) + C
• cos(x) dx = sin(x) + C
Z
•
Z
sec2 (x) dx = tan(x) + C
•
sec(x) tan(x) dx = sec(x) + C
•
sec(x) dx = ln | sec(x) + tan(x)| + C
•
Z
•
Z
Z
•
Z
•
Z
•
csc2 (x) dx = − cot(x) + C
csc(x) cot(x) dx = − csc(x) + C
Z
Z
1
dx = arctan(x) + C
x2 + 1
•
ex dx = ex + C
•
Z
√
1
dx = arcsin(x) + C
1 − x2
x
1
1
dx
=
arctan
+C
x2 + a2
a
a
ax dx =
ax
+C
ln(a)
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While the indefinite integral f (x) dx is a family of functions (all antiderivatives of f ), the definite integral
Rb
f (x) dx is a number. The definite integral can be interpreted as a difference of areas. Geometrically,
a
Z
b
f (x) dx = A1 − A2
a
where A1 is the area enclosed by the graph of the function above the x-axis (between x = a and x = b) and
A2 is the area enclosed by the graph of the function below the x-axis.
Z 1p
Example 1. Evaluate
1 − x2 dx.
0
√
Intuition: From a geometric perspective this problem√is quite simple. As f (x) = 1 − x2 ≥ 0, we need to
determine the area enclosed between the graph of y = 1 − x2 and the x-axis when 0 ≤ x ≤ 1.
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Calculus II Resources
As the graph of y =
√
Integration Techniques
1 − x2 is the top half of a circle of radius 1 centered at the origin, we have
1
Z
p
1 − x2 dx = A
0
1
π(1)2
4
π
= .
4
=
The Fundamental Theorem of Calculus (FTOC) establishes a fantastic connection between indefinite
and definite integrals. If f is continuous on [a, b] and F is any antiderivative of f , then
b
Z
f (x) dx = F (b) − F (a).
a
Z
π/2
cos(x) dx.
Example 2. Evaluate
0
Intuition: Since
d
dx
sin(x) = cos(x), an antiderivative of cos(x) is sin(x). Apply the FTOC
Z
0
π/2
π/2
cos(x) dx = sin(x)
0
π
− sin(0)
= sin
2
=1
While it can be relatively systematic to differentiate arbitrary function it is typically much more difficult
to antidifferentiate. The purpose of this resource is to provide an intuitive review of the the most common
techniques of integration. Each section includes an overview, fully worked examples with accompanying video
solutions, a summary of key observations, a one-page handout, and practice problems with accompanying
written and video solutions.
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