Integration Methods

Why This Matters

Integration is the backbone of Unit 6 and shows up everywhere in AP Calculus—from finding areas and volumes to solving differential equations in Unit 7. You're being tested on your ability to recognize which technique fits which integral, not just whether you can mechanically execute a formula. The exam loves to present integrals that look intimidating but collapse beautifully once you identify the right approach: substitution, parts, partial fractions, or trigonometric manipulation.

Think of integration methods as a toolkit. Each technique exists because certain integral forms resist the basic power rule, and mathematicians developed specific strategies to handle them. The Fundamental Theorem of Calculus connects all of this to accumulation and net change, so every method you master here directly supports your ability to model real-world problems. Don't just memorize steps—know why each method works and when to reach for it.

---

Direct Integration: The Foundation

These methods handle the simplest cases and form the building blocks for everything else. When an integral matches a known form directly, apply the rule without overcomplicating it.

Power Rule and Constant Multiple Rule

• Power rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$—this is your first attempt on any polynomial term
• Constant multiple rule lets you factor out constants: $\int k \cdot f(x) \, dx = k \int f(x) \, dx$, simplifying coefficients before integrating
• Always add $C$ for indefinite integrals—forgetting this constant of integration costs easy points on FRQs

Inverse Trigonometric Integrals

• Recognize the forms: $\int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin(x) + C$ and $\int \frac{1}{1+x^2} \, dx = \arctan(x) + C$—these appear frequently
• Pattern matching is key: any integral resembling $\frac{1}{\sqrt{a^2 - x^2}}$ or $\frac{1}{a^2 + x^2}$ signals an inverse trig result
• Memorize the derivatives of $\arcsin$, $\arctan$, and $\text{arcsec}$—integration reverses these relationships

---

Substitution Methods: Simplifying Structure

When the integrand contains a composite function, substitution transforms it into something manageable. The chain rule in reverse—look for an "inside function" and its derivative nearby.

U-Substitution

• Core idea: Let $u = g(x)$, then $du = g'(x) \, dx$—rewrite the entire integral in terms of $u$ only
• Look for the chain rule pattern: the integrand should contain both a composite function and (a multiple of) the derivative of the inner function
• Substitute back after integrating to express your answer in terms of $x$; for definite integrals, you can convert limits to $u$-values instead

Trigonometric Substitution

• Use when you see radicals: $\sqrt{a^2 - x^2}$ suggests $x = a\sin\theta$; $\sqrt{a^2 + x^2}$ suggests $x = a\tan\theta$; $\sqrt{x^2 - a^2}$ suggests $x = a\sec\theta$
• Pythagorean identities eliminate the radical: for example, $\sqrt{a^2 - a^2\sin^2\theta} = a\cos\theta$
• Convert everything: replace $dx$, adjust limits for definite integrals, and draw a reference triangle to convert back to $x$

---

Integration by Parts: Products of Functions

When the integrand is a product of two different function types, parts separates them strategically. Based on the product rule in reverse: $\int u \, dv = uv - \int v \, du$.

Integration by Parts

• Formula: $\int u \, dv = uv - \int v \, du$—choose $u$ to be the function that simplifies when differentiated, and $dv$ to be easily integrable
• LIATE priority helps with choosing $u$: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential—earlier categories typically become $u$
• Repeat if necessary: integrals like $\int x^2 e^x \, dx$ require applying parts multiple times; watch for cyclic patterns with trig-exponential products

---

Rational Function Techniques: Breaking Down Fractions

Rational functions (polynomial over polynomial) require algebraic manipulation before integration. The goal is to decompose complexity into integrable pieces.

Polynomial Long Division

• When to use: if the degree of the numerator is greater than or equal to the degree of the denominator, divide first
• Result: a polynomial (easy to integrate) plus a proper fraction where the numerator's degree is less than the denominator's
• Then proceed with partial fractions or direct integration on the remaining proper fraction

Partial Fraction Decomposition

• Purpose: break a proper rational function into a sum of simpler fractions like $\frac{A}{x-a}$ or $\frac{Bx + C}{x^2 + k}$
• Factor the denominator completely into linear and irreducible quadratic factors; set up unknown constants for each factor
• Solve for constants by clearing denominators and matching coefficients or substituting strategic $x$-values

---

Trigonometric Integrals: Identity-Based Simplification

Integrands with powers of sine, cosine, tangent, or secant often require identity manipulation. Pythagorean identities and power-reducing formulas are your primary tools.

Trigonometric Integrals

• Odd powers of sin or cos: save one factor for $du$ and convert the rest using $\sin^2 x + \cos^2 x = 1$
• Even powers: use power-reducing identities like $\sin^2 x = \frac{1 - \cos(2x)}{2}$ and $\cos^2 x = \frac{1 + \cos(2x)}{2}$
• Tangent-secant integrals: save $\sec^2 x$ for $du$ when you have even secant powers, or $\sec x \tan x$ when you have odd tangent powers

---

Improper Integrals: Handling Infinity

When limits extend to infinity or the integrand has discontinuities, standard techniques need modification. Replace problematic bounds with limits and evaluate convergence.

Improper Integrals

• Infinite limits: evaluate $\int_a^{\infty} f(x) \, dx$ as $\lim_{t \to \infty} \int_a^t f(x) \, dx$—if the limit exists, the integral converges
• Discontinuous integrands: split the integral at the discontinuity and take one-sided limits approaching the problem point
• Convergence tests: compare to known convergent/divergent integrals; $\int_1^{\infty} \frac{1}{x^p} \, dx$ converges for $p > 1$ and diverges for $p \leq 1$

---

Quick Reference Table

---

Self-Check Questions

1. You encounter $\int \frac{x^3 + 2x}{x^2 + 1} \, dx$. Which two techniques must you apply, and in what order?

2. Compare u-substitution and integration by parts: what structural feature in the integrand tells you which method to use?

3. Which trigonometric substitution would you use for $\int \frac{1}{\sqrt{9 - x^2}} \, dx$, and what identity makes the radical disappear?

4. An FRQ asks whether $\int_1^{\infty} \frac{1}{x^{3/2}} \, dx$ converges or diverges. What's your answer and how do you justify it?

5. You're integrating $\int \frac{3x + 5}{(x-1)(x+2)} \, dx$. Set up the partial fraction decomposition and explain why this form works.