Integration Techniques

Review: Basic Integration

Power rule: ∫xⁿ dx = x^(n+1)/(n+1) + C (n ≠ -1)

Basic rules:

∫sin(x) dx = -cos(x) + C
∫cos(x) dx = sin(x) + C
∫eˣ dx = eˣ + C
∫(1/x) dx = ln|x| + C

Need techniques for more complex integrals

U-Substitution

Method: Change of variables to simplify integral

Formula: ∫f(g(x))g'(x) dx = ∫f(u) du where u = g(x)

Steps:

1. Choose u (usually inner function)
2. Find du = u'(x) dx
3. Substitute to get integral in u
4. Integrate
5. Substitute back to x

Example 1: U-Substitution

∫2x(x² + 1)³ dx

Let u = x² + 1

du = 2x dx

Substitute:

= ∫u³ du
= u⁴/4 + C
= (x² + 1)⁴/4 + C

Example 2: Adjusting Constants

∫x cos(x²) dx

Let u = x²

du = 2x dx
x dx = du/2

Substitute:

= ∫cos(u) · (du/2)
= (1/2)∫cos(u) du
= (1/2)sin(u) + C
= (1/2)sin(x²) + C

U-Substitution for Definite Integrals

Two approaches:

1. Change limits to u-values
2. Substitute back to x, use original limits

Example: Definite Integral

∫[0 to 1] x√(1 + x²) dx

Let u = 1 + x², du = 2x dx

Change limits:

• x = 0: u = 1
• x = 1: u = 2

Integrate:

= (1/2)∫[1 to 2] √u du
= (1/2)∫[1 to 2] u^(1/2) du
= (1/2) · (2/3)u^(3/2)|₁²
= (1/3)[2^(3/2) - 1]
= (1/3)[2√2 - 1]

Integration by Parts

Formula: ∫u dv = uv - ∫v du

Reverse of product rule

Choosing u and dv:

• LIATE rule: Logarithm, Inverse trig, Algebraic, Trig, Exponential
• Choose u from left, dv from right

Example 1: By Parts

∫x eˣ dx

Choose:

• u = x → du = dx
• dv = eˣ dx → v = eˣ

Apply formula:

= x·eˣ - ∫eˣ dx
= x·eˣ - eˣ + C
= eˣ(x - 1) + C

Example 2: Logarithm

∫ln(x) dx

Choose:

• u = ln(x) → du = (1/x) dx
• dv = dx → v = x

Apply:

= x·ln(x) - ∫x·(1/x) dx
= x·ln(x) - ∫1 dx
= x·ln(x) - x + C

Repeated Integration by Parts

Sometimes need to apply by parts multiple times

Example: xe^(-x)

∫x² eˣ dx

First application:

• u = x² → du = 2x dx
• dv = eˣ dx → v = eˣ

= x²eˣ - ∫2x eˣ dx

Second application on ∫2x eˣ dx:

• u = 2x → du = 2 dx
• dv = eˣ dx → v = eˣ

= x²eˣ - [2x eˣ - ∫2eˣ dx]
= x²eˣ - 2x eˣ + 2eˣ + C
= eˣ(x² - 2x + 2) + C

Trigonometric Integrals

Powers of sin and cos

Strategies:

• Odd power: Save one, use identity sin²x + cos²x = 1
• Even powers: Use half-angle formulas

Example: Odd Power

∫sin³(x) dx

Rewrite:

= ∫sin²(x)·sin(x) dx
= ∫(1 - cos²(x))·sin(x) dx

Let u = cos(x), du = -sin(x) dx:

= -∫(1 - u²) du
= -[u - u³/3] + C
= -cos(x) + cos³(x)/3 + C

Example: Even Power

∫cos²(x) dx

Use identity: cos²(x) = (1 + cos(2x))/2

= ∫(1 + cos(2x))/2 dx
= (1/2)∫1 dx + (1/2)∫cos(2x) dx
= x/2 + sin(2x)/4 + C

Trigonometric Substitution

For expressions with √(a² - x²), √(a² + x²), √(x² - a²)

Substitutions:

• √(a² - x²): x = a sin(θ)
• √(a² + x²): x = a tan(θ)
• √(x² - a²): x = a sec(θ)

Example: Trig Substitution

∫√(4 - x²) dx

Let x = 2sin(θ), dx = 2cos(θ) dθ

Substitute:

= ∫√(4 - 4sin²θ) · 2cos(θ) dθ
= ∫2cos(θ) · 2cos(θ) dθ
= 4∫cos²(θ) dθ
= 4 · [θ/2 + sin(2θ)/4] + C
= 2θ + sin(2θ) + C

Back to x:

θ = arcsin(x/2)
sin(2θ) = 2sin(θ)cos(θ) = 2(x/2)√(1-x²/4) = (x/2)√(4-x²)

= 2arcsin(x/2) + (x/2)√(4-x²) + C

Partial Fractions

Decompose rational function into simpler fractions

For: ∫P(x)/Q(x) dx where degree(P) < degree(Q)

Factor denominator, set up partial fractions

Example: Partial Fractions

∫(5x - 1)/(x² - x - 2) dx

Factor: x² - x - 2 = (x - 2)(x + 1)

Decompose:

(5x - 1)/[(x - 2)(x + 1)] = A/(x - 2) + B/(x + 1)

Solve for A, B:

5x - 1 = A(x + 1) + B(x - 2)

x = 2: 9 = 3A → A = 3
x = -1: -6 = -3B → B = 2

Integrate:

= ∫[3/(x - 2) + 2/(x + 1)] dx
= 3ln|x - 2| + 2ln|x + 1| + C

Integration Tables

For complex integrals, use tables or CAS

Common forms catalogued with solutions

Modern approach: Computer algebra systems (Wolfram Alpha, Mathematica)

Choosing the Right Technique

1. Simplify first: Algebra might help

2. Check basic rules: Power rule, standard integrals

3. U-substitution: Look for composition f(g(x))g'(x)

4. By parts: Products of different types (LIATE)

5. Trig identities: Powers of trig functions

6. Trig substitution: Square roots with a² ± x²

7. Partial fractions: Rational functions

8. Technology: When all else fails!

Practice