Trigonometric Substitution
Integrands involving the sum or difference of squares may sometimes be evaluated by
using trigonometric substitution. The majority of integrals in this section have the following
forms as a part of the integrand.
x 2 a2 ,
a2 x2
Numerical Integration
Left-End Point Approximation:
b
Z
f (x) dx
a
n1
X
f (xk )xk =
k=0
ba
[y0 + y1 + +yn1 ]
n
Right-End Point Approximation:
Z
b
f (x) dx
a
n
X
f (xk )xk =
k=1
ba
[y1 + y1 + +yn ]
n
Midpoint Approximation:
Z
b
f (x) dx
a
n
X
f (mk )xk
Trigonometric Integrals
Powers of Sine and Cosine: Integrals of the form
Z
sinm x cosn x dx
If either m or n is odd, use u-substitution. If m and n are even, use the following power reducing
formulas to reduce the powers on the sine and cosine functions.
Integration by Parts
Recall the product rule for differentiation: If f and g are differential functions then
d
[f (x) g(x)] = f (x) g 0 (x) + g(x) f 0 (x)
dx
which in the notation for indefinite integrals is equivalent to
Z
[f (x) g 0 (x) + g(x) f 0 (x)]
u-Substitution
Recall the chain rule: If f and g are differential functions and F = f g = f (g(x), then F
is differentiable and
F 0 (x) = (f g)0 (x) = f 0 (g(x) g 0 (x)
Examples: Differentiate the following functions:
1. F (x) = (3x2 + 2x 1)6
2. F (x) = c
Mathematics 280
Chapter 11: Sequences and Series
Comparison Test
Direct Comparison Test: Suppose
an and
n=1
0 an bn for all n 1. Then
X
X
bn converges =
n=1
X
X
bn are positive term series and
n=1
X
an converges
n=1
an diverges =
n=1
X
bn diverges
n=1
Loo
Infinite Series
Definition: Consider the infinite series
X
an = a1 + a + a2 + + an +
n=1
We denote the nth partial sum as
Sn =
n
X
ak = a1 + a + a2 + + an
k=1
If the sequence of partial sums cfw_Sn
n=1 converges to a limit S, that is lim Sn = S
n
X
an c
Mathematics 280
Chapter 11: Sequences and Series
Integral Test
Theorem: Let
X
an be a series with positive terms and let f (x) be the function
n=1
that results when n is replaced by x in the formula for f (x). If f is a decreasing and
Z
X
f (x) dx both c
Improper Integrals
Recall the definite Integral
Rb
a
f (x)dx with the following constraints:
1. the interval [a, b] has finite length,
2. f does not have an infinite discontinuity on [a, b].
Definition: A function f is said to have an infinite discontinui
Analytic Geometry and Calculus II
Section 8024
Grossmont College
Spring 2016
Instructor: Michael J. Lines
Phone: (619) 644-7320
E-mail: michael.lines@gcccd.edu.
Office: 31-383A
Office Hours Monday /Wednesday: 2:00 4:00 p.m.
Tuesday/Thursday: 8:30 - 9:00 a