Integration Techniques

Chapter 8.1 - 8.6

Besides the rules of integration that we've already discussed, there are more methods of integration that you can use. In this section, we will cover some of these rules for integration, which will help you work with more difficult integrals.

Chapter 8.1

Basic Integration Rules (A Review)

Procedures for Fitting Integrals to Basic Rules:

Expand the numerator (multiply it out!)
Separate a constant number from the rest of the integral
Complete the square (when using arc function formulas)
Divide improper rational functions
Add and subtract terms in the numerator
Use trigonometric identities and formulas
Multiply and divide by Pythagorean conjugate

Pythagorean conjugates:

Example: (1 - sin x) and (1 + sin x) are Pythagorean conjugates of each other

Chapter 8.2

Integration by Parts

Theorem 8.1 - Integration by Parts:

If u and v are functions of x and have continuous derivatives, then

∫ u dv = uv - ∫ v du

Guidelines for Integration by Parts:

Try letting dv be the most complicated portion of the integrand that fits a basic integration rule. Then u will be the remaining factor(s) of the integrand.
Try letting u be the portion of the integrand whose derivative is a function simpler than u. Then dv will be the remaining factor(s) of the integrand.

Summary of Common Integrals Using Integration by Parts:

1. For integrals of the forms:

∫ (x^n) (e^ax)dx
∫ (x^n) sin ax dx
∫ (x^n) cos ax dx

let u = x^n and let dv = (e^ax)dx, sin ax dx, or cos ax dx

2. For integrals of the forms:

∫ (x^n) ln x dx
∫ (x^n) arcsin ax dx
∫ (x^n) arctan ax dx

let u = ln x dx, arcsin ax dx, or arctan ax dx and let dv = x^n dx

3. For integrals of the forms:

∫ (e^ax) sin bx dx
∫ (e^ax) cos bx dx

let u = sin bx or cos bx and let dv = e^ax dx

Tabular Method:

Works well for integrals of the forms ∫ (x^n) sin ax dx, ∫ (x^n) cos ax dx, and ∫ (x^n) (e^ax)dx.

Example:

Chapter 8.3

Trigonometric Integrals

Guidelines for Evaluating Integrals Involving Sine and Cosine:

1. If the power of the sine is odd and positive, save one sine factor and convert the remaining factors to cosines. Then, expand and integrate.

∫ sin^(2k + 1) x cos^n x dx

= ∫ (sin^2 x)^k cos^n x sin x dx

= ∫ (1 - cos^2 x)^k cos^n x sin x dx

2. If the power of the cosine is odd and positive, save one cosine factor and convert the remaining factors to sines. then expand and integrate.

∫ sin^m x cos^(2k + 1) x dx

= ∫ sin^m x (cos^2 x)^k cos x dx

= ∫ sin^m x (1 - sin^2 x)^k cos x dx

3. If the power of both the sine and the cosine are even and nonnegative, make repeated use of the identities

sin^2 x = (1 - cos 2x)/ 2

cos^2 x = (1 + cos 2x)/ 2

to convert the integrand to odd powers of the cosine. Then proceed as in guideline 2.

Wallis's Formulas:

1. If n is odd (n ≥ 3), then

(∫ π/2 -> 0) cos^n x dx

= (2/3)(4/5)(6/7)… ((n - 1)/n)

2. If n is even (n ≥ 2), then

(∫ π/2 -> 0) cos^n x dx

= (1/2)(3/4)(5/6)… (π/2)((n - 1)/n)

Guidelines for Evaluating Integrals Involving Secant and Tangent:

1. If the power of the secant is even and positive, save a secant-squared factor and convert the remaining factors to tangents. Then expand and integrate.

∫ sec^2k x tan^n x dx

= ∫ (sec^2 x)^(k - 1) tan^n x sec^2 x dx

= ∫ (1 + tan^2 x)^(k - 1) tan^n x sec^2 x dx

2. If the power of the tangent is odd and positive, save a secant tangent factor and convert the remaining factors to secants. Then expand and integrate.

∫ sec^m x tan^(2k + 1) x dx

= ∫ sec^(m - 1) x (tan^2 x)^k sec x tan x dx

= ∫ sec^(m - 1) x (sec^2 x - 1)^k sec x tan x dx

3. If there are no secant factors and the power of the tangent is even and positive, convert a tangent-squared factor to a secant-squared factor, then expand and repeat if necessary.

∫ tan^n x dx

= ∫ tan^(n - 2) x (tan^2 x) dx

= ∫ tan^(n - 2) x (sec^2 x - 1) dx

4. If the integral is of the form ∫ sec^m d dx, where m is odd and positive, use integration by parts. (Split the secant into two factors, u and dv)

5. If none of the first four guidelines apples, try converting into sines and cosines.

Sine-Cosine Products with Different Angles:

cos mx cos nx = 0.5(cos [(m - n)x] + cos [(m + n)x])

sin mx sin nx = 0.5(cos [(m - n)x] - cos [(m + n)x])

sin mx cos nx = 0.5(sin [(m - n)x] + sin [(m + n)x])

cos mx sin nx = 0.5(sin [(m - n)x] - sin [(m + n)x])

Chapter 8.4

Trigonometric Substitution

The objective with trigonometric substitution is to eliminate the radical in the integrand.

Trigonometric Substitution (a > 0):

1. For integrals involving √(a^2 - u^2), let u = a sin θ

Then √(a^2 - u^2) = a cos √(a^2 - u^2),

where -π/2 ≤ θ ≤ π/2.

2. For integrals involving √(a^2 + u^2), let u = a tan θ

Then √(a^2 + u^2) = a sec θ,

where -π/2 ≤ θ ≤ π/2.

3. For integrals involving √(u^2 - a^2), let u = a sec θ

Then √(u^2 - a^2) = ± a tan θ,

where 0 ≤ θ < π/2 or π/2 < θ < π.

Use the positive value if u > a and the negative value if u < -a.

Theorem 8.2 - Special Integration Formulas (a > 0):

1. ∫ √(a^2 - u^2) du

= 0.5[u√(a^2 - u^2) + a^2 arcsin (u/a)] + C

2. ∫ √(u^2 - a^2) du

= 0.5[u√(u^2 - a^2) - a^2 ln |u + √(u^2 - a^2)|] + C

(u > a)

3. ∫ √(u^2 + a^2) du

= 0.5[u√(u^2 + a^2) + a^2 ln |u + √(u^2 + a^2)|] + C

Chapter 8.5

Partial Fractions

Decomposition of N(x)/D(x) into Partial Fractions:

1. Divide if improper: If N(x)/D(x) is an improper fraction (that is, if the degree if the numerator is greater than or equal to the degree of the denominator), divide the denominator into the numerator to obtain

N(x)/D(x) = (a polynomial) + (N-^1 (x)/D(x))

where the degree of N-^1 (x) is less than the degree of D(x). Then apply steps 2, 3, and 4 to the proper rational expression N-^1 (x)/D(x).

2. Factor denominator: Completely factor the denominator into factors of the forms

(px + q)^m and (ax^2 + bx + c)^n

where ax^2 + bx + c is irreducible.

3. Linear factors: For each factor of the form (px + q)^m, the partial fraction decomposition must include the following sum of m fractions.

(A-^1 /(px + q)) + (A-^2 /(px + q)^2) + … + (A-^m /(px + q)^m)

4. Quadratic Factors: For each factor of the form (ax^2 + bx + c)^n, the partial fraction decomposition must include the following sum of n fractions.

((B-^1 x + C-^1)/(ax^2 + bx + c)) + ((B-^2 x + C-^2)/(ax^2 + bx + c)^2) + … + ((B-^n x + C-^n)/(ax^2 + bx + c)^n)

Guidelines for Solving the Basic Equation:

Linear Factors:

Substitute the roots of the distinct linear factors into the basic equation.
For repeated linear factors, use the coefficients determined in Guideline 1 to rewrite the basic equation. Then substitute other convenient values of x and solve for the remaining coefficients.

Quadratic Factors:

Expand the basic equation.
Collect terms according to powers of x.
Equate the coefficients of like powers to obtain a system of linear equations involving A, B, C, and so on.
Solve the system of linear equations.

Chapter 8.6

Integration by Tables and Other Integration Techniques

integration by tables - uses tables of formulas of common integrals

reduction formulas - formulas that reduce a given integral to the sum of a function and a simpler integral; have the form ∫ f(x) dx = g(x) + ∫ h(x) dx

Substitution for Rational Functions of Sine and Cosine:

For integrals involving rational functions of sine and cosine, the substitution

u = sin x /(1 + cos x) = tan x/2

yields:

cos x = (1 - u^2)/(1 + u^2)

sin x = 2u/(1 + u^2)

dx = 2du/(1 + u^2)

The PDF document below contains integral tables that you might find useful. Credits go to http://integral-table.com/

Chapter 8.7

Indeterminate Forms and L'Hôpital's Rule

Indeterminate forms do not guarantee that a limit exists, nor do they indicate what the limit is, if one does exist

Examples:

0/0
∞/∞
∞ * 0
∞ - ∞
∞^0
0^0
1^∞

Theorem 8.3 - The Extended Mean Value Theorem:

If f and g are differentiable on an open interval (a, b) and continuous on [a, b] such that g′(x) ≠ 0 for any x in (a, b), then there exists a point c in (a, b) such that

(f′(c)/g′(c)) = (f(b) - f(a))/(g(b) - g(a))

Theorem 8.4 - L'Hôpital's Rule:

Let f and g be functions that are differentiable on an open interval (a, b) containing c, except possibly at c itself. Assume that g′(x) ≠ 0 for all x in (a, b), except possibly at c itself. If the limit of f(x)/g(x) as x approaches c produces the indeterminate form 0/0 (or another indeterminate form), then

lim (-^- (x -> c)) f(x)/g(x)

= lim (-^- (x -> c)) f′(x)/g′(x)

provided the limit on the right exists (or is infinite). This result also applies if the limit of f(x)/g(x) as x approaches c produces any one of the indeterminate forms

∞/∞
-∞/∞
∞/-∞
-∞/-∞

NOTE: People occasionally use the L'Hôpital's Rule incorrectly by applying the Quotient Rule to f(x)/g(x). Be sure you see that the rule involves f′(x)/g′(x), not the derivative of f(x)/g(x).

Another Form of L'Hôpital's Rule:

If lim (-^- (x -> (-)∞)) f(x)/g(x) = 0/0 or ∞/∞, then

lim (-^- (x -> ∞)) f(x)/g(x)

= lim (-^- (x -> ∞)) f′(x)/g′(x)

provided the limit on the right exists.

Chapter 8.8

Improper Integrals

improper integrals - one or both of the limits of integration are infinite; or f has a finite number of infinite discontinuities in the integral [a, b]

infinite discontinuity - lim (-^- (x -> c)) f(x) = ∞ or -∞

Definition of Improper Integral with Infinite Integration Limits:

1. If f is continuous on the interval [a, ∞), then

(∫ a-> ∞) f(x) dx

= lim (-^- (x -> ∞)) (∫ a-> b) f(x) dx

2. If f is continuous on the interval (-∞, b], then

(∫ -∞ -> b) f(x) dx

= lim (-^- (x -> -∞)) (∫ a-> b) f(x) dx

3. If f is continuous on the interval (-∞, ∞), then

(∫ -∞ -> ∞) f(x) dx

= (∫ -∞ -> c) f(x) dx + (∫ c-> ∞) f(x) dx

where c is any real number.

In the first two cases, the improper integral converges if the limit exists — otherwise, the improper integral diverges. In the third case, the improper integral on the left diverges if either of the improper integrals on the right diverges.

Definition of Improper Integrals with Infinite Discontinuities:

1. If f is continuous on the interval [a, b) and has an infinite discontinuity at b, then

(∫ a-> b) f(x) dx

= lim (-^- (c -> b- )) (∫ a-> c) f(x) dx

2. If f is continuous on the interval (a, b] and has an infinite discontinuity at a, then

(∫ a-> b) f(x) dx

= lim (-^- (c -> a+ )) (∫ c-> b) f(x) dx

3. If f is continuous on the interval [a, b], except for some c in (a, b) at which f has an infinite discontinuity, then

(∫ a-> b) f(x) dx

= (∫ a-> c) f(x) dx + (∫ c-> b) f(x) dx

Theorem 8.5 - A Special Type of Improper Integral:

(∫ 1-> ∞) dx/x^p =

1/(p - 1), if p > 1

diverges, if p ≤ 1