Maths I — Week 7: Advanced Integration

Short description. Advanced integration involves reversing the chain rule and product rule. These techniques allow us to calculate areas and accumulated quantities for complex mathematical models.

---

1. Core Concept

• Substitution is the Inverse Chain Rule.
• Integration by Parts is the Inverse Product Rule.
• Partial Fractions is the Inverse Common Denominator.

---

2. Pattern A — The Substitution Protocol (u-substitution)

Abstract Strategy

1. [Identification]: Identify the "inner" function $g(x)$ whose derivative is sitting outside.
2. [Variable Swap]: Let $u = g(x)$, then $du = g'(x)\,dx$.
3. [Re-normalization]: Rewrite the entire integral in terms of $u$. Solve and finally revert to $x$.

Procedure

• Step 1: Choose $u$ (usually the part under a power or inside a trig function).
• Step 2: Calculate $du$ and solve for $dx$ if needed.
• Step 3: Substitute and integrate using the Power Rule or Transcendental rules.

Question: Compute $\int 2x(x^2 + 1)^5 \, dx$

• Step 1: Let $u = x^2 + 1$.
• Step 2: $du = 2x \, dx$.
• Step 3: $\int u^5 \, du = \frac{u^6}{6} + C$.
• Answer: $\frac{(x^2+1)^6}{6} + C$

---

3. Pattern B — Integration by Parts (ILATE Rule)

Abstract Strategy

1. [Priority Check]: Determine which function is $u$ using the ILATE priority hierarchy:
   • I: Inverse Trig
   • L: Logarithmic
   • A: Algebraic ($x^n$)
   • T: Trigonometric
   • E: Exponential
2. [Derivative/Integral Map]: Find $du$ by differentiating $u$ and find $v$ by integrating $dv$.
3. [Apply Identity]: Plug into $uv - \int v\,du$.

Procedure

• Step 1: Assign $u$ and $dv$. (Algebraic usually beats Trig/Exp for $u$).
• Step 2: Compute $du = u'dx$ and $v = \int dv$.
• Step 3: Plug in and solve the remaining integral.

Question: Compute $\int x \cdot \ln(x) \, dx$

• Step 1: $u = \ln(x)$ (Log before Algebraic). $dv = x \, dx$.
• Step 2: $du = \frac{1}{x} dx$, $v = \frac{x^2}{2}$.
• Step 3: $\frac{x^2}{2} \ln(x) - \int \frac{x^2}{2} \cdot \frac{1}{x} \, dx$.
• Answer: $\frac{x^2}{2}\ln(x) - \frac{x^2}{4} + C$

---

4. Pattern C — Partial Fraction Decomposition

Abstract Solution (Strategy)

1. [Factor denominator]: Fully factor $Q(x)$ into linear/quadratic factors.
2. [Set up partial fractions]: For each distinct linear factor $(x - r)$, write a term $\frac{A}{x-r}$. For repeated factors $(x-r)^2$, write $\frac{A}{x-r} + \frac{B}{(x-r)^2}$.
3. [Solve for constants]: Multiply both sides by $Q(x)$ and equate coefficients (or plug in roots).
4. [Integrate each term]: Use $\int \frac{1}{x-r}\,dx = \ln|x-r| + C$.

Procedure

• Step 1: Verify $\deg P < \deg Q$ (if not, perform polynomial long division first).
• Step 2: Decompose the fraction.
• Step 3: Find constants by plugging in the roots of $Q(x)$.
• Step 4: Integrate each simple fraction.

$\int \frac{1}{(x-1)(x+2)}\,dx$

• Step 1: $\frac{1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$.
• Step 2: Multiply: $1 = A(x+2) + B(x-1)$.
• Step 3: At $x=1$: $1 = 3A \implies A = 1/3$. At $x=-2$: $1 = -3B \implies B = -1/3$.
• Step 4: $\int \frac{1/3}{x-1}\,dx - \int \frac{1/3}{x+2}\,dx = \frac{1}{3}\ln|x-1| - \frac{1}{3}\ln|x+2| + C$.
• Result: $\frac{1}{3}\ln\left|\frac{x-1}{x+2}\right| + C$.

---

5. Definite Integration with u-Substitution (Limit Change)

$\int_0^1 2x(x^2+1)^5\,dx$

• Let $u = x^2+1$. When $x=0$, $u=1$. When $x=1$, $u=2$.
• $\int_1^2 u^5\,du = \left[\frac{u^6}{6}\right]_1^2 = \frac{64}{6} - \frac{1}{6} = \frac{63}{6} = \frac{21}{2}$.

---

6. Common Mistakes

---

7. Flashcards

<Flashcard front="What is the ILATE rule priority order?" back="Inverse Trig > Logarithmic > Algebraic > Trigonometric > Exponential." /> <Flashcard front="When do you use Partial Fraction decomposition?" back="When integrating a rational function P(x)/Q(x) where the degree of P is less than the degree of Q." /> <Flashcard front="State the Integration by Parts formula." back="∫u dv = uv - ∫v du." /> <Flashcard front="How do you handle limits in u-substitution for definite integrals?" back="Convert the limits: new lower limit = g(a), new upper limit = g(b). Don't substitute back to x." /> <Flashcard front="For partial fractions, what do you do if the numerator degree ≥ denominator degree?" back="Perform polynomial long division first to reduce to a proper fraction before decomposing." />

---

8. Practice Targets

• Compute $\int x^2 e^x\,dx$ using the Tabular Method (repeated Integration by Parts).
• Decompose $\frac{2x+1}{x^2+3x+2}$ into partial fractions and integrate.
• Evaluate $\int_0^{\pi/2} x\cos x\,dx$ using IBP.
• Attempt Integration questions from Graded Assignment 7 (Questions 1--5).

---

9. Connections