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Definitions and Theorems
Definitions
• The definite integral of a continuous function f (x) on an interval [a, b] is
Z b
n
X
f (x) dx = lim
f (ci )∆x,
∆x→0
a
i=1
where n is the number of partitions, ci is a point in the ith partition, and ∆x is the width of
the ith partition. (The definite integral is the limit of Riemann Sums.)
• The indefinite integral of a continuous function f (x),
Z
d
f (x) dx = F (x) + c, where dx
F (x) = f (x)
is the set of all Antiderivatives of f (x).
Theorems
• Mean Value Theorem for Integrals
If f (x) is continuous on the interval [a, b], then there exists c ∈ [a, b] such that
Z b
1
f (x) dx = Average of f (x) on [a, b].
f (c) = b−a
a
• Fundamental Theorem of Calculus
If f (x) is continuous on the interval [a, b] and
Z
F (x) =
x
f (t) dt,
a
d
then dx
F (x) = f (x) for a ≤ x ≤ b.
If f (x) is continuous on the interval [a, b] and F (x) is an Antiderivative of f (x) then
Z b
f (x) dx = F (b) − F (a).
a
Key Integral Formulas to Know
Z
1.
Z
kf (x) dx = k
f (x) dx
Z
Z
f (x) ± g(x) dx =
2.
Z
Z
f (x) dx ±
g(x) dx
a
3.
f (x) dx = 0
a
Z
a
Z
f (x) dx = −
4.
f (x) dx
b
Z
5.
a
c
Z
f (x) dx =
a
b
b
Z
f (x) dx +
a
c
f (x) dx
b
Z
6.
Z
k dx = kx + c
Z
7.
12.
csc2 (x) dx = − cot(x) + c
n+1
xn dx = xn+1 + c
Z
csc(x) cot(x) dx = − csc(x) + c
13.
Z
8.
cos(x) dx = sin(x) + c
Z
14.
Z
ex dx = ex + c
sin(x) dx = − cos(x) + c
9.
Z
10.
a
+c
ax dx = ln(a)
Z
1
x dx = ln(x) + c
15.
2
sec (x) dx = tan(x) + c
Z
11.
Z
16.
sec(x) tan(x) dx = sec(x) + c
x
Other Formulas to Recognize
Z
1.
Z
2.
f 0 (g(x))g 0 (x) dx = f (g(x)) + c
0
f (x)g (x) dx = f (x)g(x) −
Z
g(x)f 0 (x) dx