Integration Strategy

Contents

The best approach to integration is a systematic decision based on the form of the integrand. Here are two practical guides to analyze an indefinite or definite integral and pick the right technique (or combination). Each provide a structured, decision process for selecting the most effective method to evaluate an integral. The goal is to help students recognize patterns, identify dominant structures, and confidently choose the correct technique.

Choosing the Right Integration Technique – Strategy 1

Start with Structural Recognition

Before choosing a method, identify the integrand’s dominant structure:

Composite functions (a function inside another)
Rational functions (polynomials in numerator and denominator)
Products of unlike functions (e.g., polynomial × exponential)
Trigonometric expressions (powers, products, identities)
Radicals involving quadratics
Standard forms (common integrals)

This recognition guides the technique selection.

Technique Selection Flow

U-Substitution

Use when the integrand contains a function and its derivative.

Look for expressions like (f‘(x)g(f(x))).
Common in composite functions

Examples

$\int{x\sqrt{x^2+1}dx}$

$\int{e^{3x}}dx$

$\int{\sin \left( 1-x \right) \left( 2-\cos \left( 1-x \right) \right)}^4dx$

where 𝑢 = 2−cos⁡(1−𝑥), 𝑑⁢𝑢 =−sin⁡(1−𝑥)⁢ 𝑑⁢𝑥 ⁢⇒ ⁢sin⁡(1−𝑥)⁢𝑑⁢𝑥 = −𝑑⁢𝑢

Partial Fractions

Use for rational functions.

P(x)/Q(x) where P and Q are polynomials.
Apply when the denominator factors into linear or quadratic terms.

Examples

$\int{\frac{1}{x^2-1}dx}$

$\int{\frac{3x+5}{x^2+4x+3}dx}$

Integration by Parts

Use for products of unlike functions.

Apply the LIATE rule to choose (u):

Logarithmic
Inverse trigonometric
Algebraic
Trigonometric
Exponential

Examples

$\int{xe^xdx}$

$\int{\ln \left( x \right) dx}$

Trigonometric Techniques

Different trigonometric patterns require different approaches.

Powers of Sine and Cosine
- If one power is odd: peel off one factor and use substitution.
- If both are even: use half-angle identities.
Tangent and Secant
- If secant power is even: save a sec²(x) for substitution.
- If tangent power is odd: save tan(x)sec²(x).
Products and Identities
- Use product-to-sum or angle identities when needed.

Trigonometric Substitution

Use for integrals involving radicals with quadratics and match one of the following forms.

$\sqrt{a^2-x^2}\,\,\Longrightarrow x\,\,=\,\,a\sin \left( \theta \right)$

$\sqrt{a^2+x^2}\,\,\Longrightarrow x\,\,=\,\,a\tan \left( \theta \right)$

$\sqrt{x^2-a^2}\,\,\Longrightarrow x\,\,=\,\,a\sec \left( \theta \right)$

Standard Table Integrals

Use when the integrand matches a known pattern.

Examples

$\int{e^{ax}}dx$

$\int{\frac{1}{x^2+a^2}dx}$

$\int{\sec \left( x \right) dx}$

When No Method Is Obvious

Apply algebraic manipulation:

Complete the square.
Factor expressions.
Use polynomial long division.
Add or subtract helpful terms.

These often reveal hidden structures.

A Practical Decision Tree

Is a derivative present? ⟶ U-substitution.
Is it rational? ⟶ Partial fractions.
Is it a product of unlike functions? ⟶ Integration by parts.
Is it trigonometric? ⟶ Identify the trigonometric pattern.
Is there a radical with a quadratic? ⟶ Trigonometric substitution.
Does it match a standard form? ⟶ Table integrals.
Still stuck? ⟶ Algebraic manipulation.

Choosing the Right Integration Technique – Strategy 2

Simplify the Integrand

Before choosing any advanced technique:

Algebraic simplification: Expand, factor, combine fractions, rationalize, complete the
square, multiply by conjugate, etc.
Trigonometric simplification: Use Pythagorean, double-angle, half-angle, or power-reduction
identities.
Substitution check: Sometimes a simple rewrite makes u-substitution obvious.

If you can simplify to a standard table integral (∫ e^x, ∫ 1/x, ∫ sin x, ∫ sec²(x), etc.), stop here.

Decision Tree by Form of Integrand

Rational Functions (polynomial / polynomial)
- Degree of numerator ≥ degree of denominator? ⟶ Polynomial long division first.
- After division (or if already proper):
  - Denominator factors into linear/quadratic terms? ⟶ Partial Fraction Decomposition.
  - Irreducible quadratics in denominator? May lead to arctan after integration.
- Special case: Repeated linear factors or quadratic factors ⟶ still partial fractions, but with higher powers.
Products of Different Types of Functions (e.g., xe^x, In(x) sin(x), x²cos(x))
- Use Integration by Parts (∫ u dv = uv — ∫ v du).
- LIATE rule to choose u (helps decide order):
  - Logarithmic
  - Inverse trigonometric
  - Algebraic (polynomials, roots)
  - Trigonometric
  - Exponential
- Choose the one higher in LIATE as u (differentiates to something simpler). Repeat if needed (tabular method for polynomials x trig/exponential).
Composite Functions (chain rule in reverse)
- Looks like f(g(x)) • g‘(x) or close to it? ⟶ U-substitution.
- Common triggers: nested radicals, e^{something}, In^(something), (ax+b)ⁿ, etc.
- After substitution, the integral should simplify dramatically.
Powers of Trigonometric Functions
- sin^m(x)cosⁿ(x)
  - One is odd? Save one factor for du, use identity for the rest.
  - Both even? Use power-reduction / half-angle identities.
- tan^m(x) secⁿ(x):
  - n even ⟶ save sec² for du, convert rest to tangent.
  - m odd ⟶ save sec tan for du, convert rest to secant.
- Other cases (e.g., sin/cos of multiple angles) trigonometric identities first.
Square Roots Involving Quadratics ( √(a² – x²), √(x² – a²), √(x² + a²) )
- Trigonometric Substitution is usually best
  - √(a² – x²) ⟶ x = asin(θ) or acos(θ)
  - √(x² – a²) ⟶ x = asec(θ)
  - √(x² + a²) ⟶ x = atan(θ)
- Sometimes complete the square + u-substitution first, or recognize it leads to inverse trigonometric / log forms (standard table after substitution).
Other Common Cases
- Inverse trigonometric or logs appearing naturally ⟶ often integration by parts.
- Hyperbolic functions or forms that suggest them ⟶ hyperbolic substitution (less common in Calculus 2).
- Definite integrals ⟶ sometimes use properties/symmetry before choosing technique, or switch to trigonometric substitution for nice limits.

Practical Workflow (Step-by-Step Analysis)

Look at the integrand: Categorize it (rational, product, composition, trig powers, sqrt quadratic, etc.).
Try simplest first: Can it be a direct table integral or simple u-substitution?
If not, apply the matching technique from the tree above.
After one technique, the result often requires another (e.g., trigonometric substitution + rational trigonometric integral + u-substitution or partial fractions).
Stuck? Try a different substitution or integration by parts with swapped u/dv.
Verify by differentiating your answer.

Quick Reference Cheat Sheet

Integrand Form	Primary Technique(s)	Key Clue
P(x)/Q(x) rational	Partial Fractions (+ division)	Factored denominator
f(x)•g(x) (different types)	Integration by Parts	LIATE rule
f(g(x))g'(x)	U-Substitution	Chain rule pattern
sin^m cosⁿ	Trigonometric identities + u-substitution	Odd power present
√(quadratic)	Trigonometric Substitution	Form a² ± x² or x² ± a²
tan^m secⁿ	Trigonometric identities + u-substitution	Even/odd powers of sec/tan
Basic (e^x, sin, 1/x, etc.)	Table / direct	Matches memorized form

References

“What is the best method to analyze an integral and determine if it should be integrated using Partial Fractions, Parts, u-substitution, Trigonometric Identities, Standard Table Integrals or Trigonometric Substitution?” COPILOT, Accessed July 21, 2026.

The featured image on this page is from the Integral Wallpaper website.