1) Introduction: Tackling Advanced Integrals 📚
Welcome to Level 5, where we dive into advanced integration techniques to handle integrals that go beyond basic methods. This topic covers integration by parts, trigonometric substitution, and partial fraction decomposition, equipping you to solve complex integrals encountered in calculus, physics, and engineering. These techniques build on your foundation, preparing you for Olympiad-level and university challenges.
We’ll cover:
- Integration by Parts: Reversing the product rule for integration.
- Trigonometric Substitution: Using trig identities to simplify integrals.
- Partial Fractions: Breaking down rational functions.
Quick Recap: Basic integration (e.g., \( \int x^n \, dx \)) is the foundation; now we tackle tougher forms.
2) Integration by Parts 🎓
Integration by parts is based on the product rule for differentiation: \( \int u \, dv = uv - \int v \, du \).
Definition 26.1: Integration by Parts
\( \int u \, dv = uv - \int v \, du \), where \( u \) and \( dv \) are chosen strategically (e.g., \( u \) simplifies on differentiation, \( dv \) is easy to integrate).
Example: \( \int x e^x \, dx \).
**Steps**: Select \( u \) and \( dv \), compute \( du \) and \( v \), apply the formula, and repeat if needed.
Example 1: Basic Integration by Parts
Evaluate \( \int x e^x \, dx \).
- Let \( u = x \), \( dv = e^x \, dx \).
- \( du = dx \), \( v = e^x \).
- \( \int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C \).
Answer: \( x e^x - e^x + C \).
Example 2: Repeated Application
Compute \( \int x^2 \sin x \, dx \).
- Let \( u = x^2 \), \( dv = \sin x \, dx \).
- \( du = 2x \, dx \), \( v = -\cos x \).
- \( \int x^2 \sin x \, dx = -x^2 \cos x - \int (-\cos x) \cdot 2x \, dx = -x^2 \cos x + 2 \int x \cos x \, dx \).
- Inner: \( u = x \), \( dv = \cos x \, dx \), \( du = dx \), \( v = \sin x \), \( = x \sin x - \int \sin x \, dx = x \sin x + \cos x \).
- Total: \( -x^2 \cos x + 2(x \sin x + \cos x) + C \).
Answer: \( -x^2 \cos x + 2x \sin x + 2 \cos x + C \).
3) Trigonometric Substitution 📐
Trigonometric substitution uses trig identities to simplify integrals involving \( \sqrt{a^2 - x^2} \), \( \sqrt{a^2 + x^2} \), or \( \sqrt{x^2 - a^2} \).
Definition 26.2: Trigonometric Substitution
- \( \sqrt{a^2 - x^2} \): \( x = a \sin \theta \), \( dx = a \cos \theta \, d\theta \). - \( \sqrt{a^2 + x^2} \): \( x = a \tan \theta \), \( dx = a \sec^2 \theta \, d\theta \). - \( \sqrt{x^2 - a^2} \): \( x = a \sec \theta \), \( dx = a \sec \theta \tan \theta \, d\theta \).
Example 3: \( \sqrt{a^2 - x^2} \)
Evaluate \( \int \frac{dx}{\sqrt{4 - x^2}} \).
- Let \( x = 2 \sin \theta \), \( dx = 2 \cos \theta \, d\theta \), \( \sqrt{4 - x^2} = \sqrt{4 - 4 \sin^2 \theta} = 2 \cos \theta \).
- \( \int \frac{2 \cos \theta \, d\theta}{2 \cos \theta} = \int 1 \, d\theta = \theta + C \).
- \( \theta = \arcsin\left(\frac{x}{2}\right) \).
Answer: \( \arcsin\left(\frac{x}{2}\right) + C \).
Example 4: \( \sqrt{x^2 + a^2} \)
Compute \( \int \frac{dx}{\sqrt{x^2 + 9}} \).
- Let \( x = 3 \tan \theta \), \( dx = 3 \sec^2 \theta \, d\theta \), \( \sqrt{x^2 + 9} = \sqrt{9 \tan^2 \theta + 9} = 3 \sec \theta \).
- \( \int \frac{3 \sec^2 \theta \, d\theta}{3 \sec \theta} = \int \sec \theta \, d\theta = \ln|\sec \theta + \tan \theta| + C \).
- \( \sec \theta = \frac{\sqrt{x^2 + 9}}{3} \), \( \tan \theta = \frac{x}{3} \).
Answer: \( \ln|\frac{\sqrt{x^2 + 9} + x}{3}| + C \).
4) Partial Fraction Decomposition 🔍
Partial fractions break down rational functions \( \frac{P(x)}{Q(x)} \) (where degree of \( P < \) degree of \( Q \)) into simpler fractions.
Definition 26.3: Partial Fractions
For \( \frac{P(x)}{(x-a)^k} \), use \( \frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \cdots + \frac{A_k}{(x-a)^k} \). For distinct factors, use \( \frac{A}{x-a} + \frac{B}{x-b} \).
Example 5: Distinct Linear Factors
Integrate \( \int \frac{1}{x^2 - 1} \, dx \).
- \( \frac{1}{(x-1)(x+1)} = \frac{A}{x-1} + \frac{B}{x+1} \).
- \( 1 = A(x+1) + B(x-1) \), set \( x = 1 \): \( 1 = 2A \), \( A = \frac{1}{2} \); \( x = -1 \): \( 1 = -2B \), \( B = -\frac{1}{2} \).
- \( \frac{1}{2} \int \frac{1}{x-1} \, dx - \frac{1}{2} \int \frac{1}{x+1} \, dx = \frac{1}{2} \ln|x-1| - \frac{1}{2} \ln|x+1| + C \).
Answer: \( \frac{1}{2} \ln \left| \frac{x-1}{x+1} \right| + C \).
Example 6: Repeated Factors
Integrate \( \int \frac{1}{x^2(x+1)} \, dx \).
- \( \frac{1}{x^2(x+1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1} \).
- \( 1 = A x (x+1) + B (x+1) + C x^2 \), solve: \( A = -1 \), \( B = 1 \), \( C = 1 \).
- \( -\int \frac{1}{x} \, dx + \int \frac{1}{x^2} \, dx + \int \frac{1}{x+1} \, dx = -\ln|x| - \frac{1}{x} + \ln|x+1| + C \).
Answer: \( \ln \left| \frac{x+1}{x} \right| - \frac{1}{x} + C \).
5) Practice Questions 🎯
Fundamental Practice Questions 🌱
Instructions: Evaluate the following integrals using the appropriate technique. 📚
\( \int x e^{2x} \, dx \) (by parts)
\( \int x^2 \cos x \, dx \) (by parts)
\( \int \frac{dx}{\sqrt{9 - x^2}} \) (trig substitution)
\( \int \frac{dx}{x^2 - 4} \) (partial fractions)
\( \int x \ln x \, dx \) (by parts)
\( \int \frac{dx}{\sqrt{x^2 + 16}} \) (trig substitution)
\( \int \frac{1}{x^2 - 9} \, dx \) (partial fractions)
\( \int x^3 e^x \, dx \) (by parts)
\( \int \frac{dx}{\sqrt{x^2 - 1}} \) (trig substitution)
\( \int \frac{1}{(x-1)(x+2)} \, dx \) (partial fractions)
\( \int \sin^{-1} x \, dx \) (by parts)
Challenging Practice Questions 🌟
Instructions: Solve these advanced integrals, possibly requiring multiple techniques. 🧠
Evaluate \( \int x^2 e^{-x} \, dx \) using integration by parts twice.
Compute \( \int \frac{x^2}{\sqrt{4 - x^2}} \, dx \) using trigonometric substitution.
Determine \( \int \frac{x^3}{(x-1)^2(x+2)} \, dx \) using partial fractions.
Approximate \( \int_0^1 x^2 \sin x \, dx \) using integration by parts and the first two terms of the series for \( \sin x \).
Solve \( \int \frac{x^3 + 1}{x^2 - 1} \, dx \) using a combination of techniques.
6) Summary & Cheat Sheet 📋
6.1) Integration by Parts
\( \int u \, dv = uv - \int v \, du \), choose \( u \) to simplify.
6.2) Trigonometric Substitution
Use \( x = a \sin \theta \), \( x = a \tan \theta \), or \( x = a \sec \theta \) for radicals.
6.3) Partial Fractions
Decompose \( \frac{P(x)}{Q(x)} \) into \( \frac{A}{x-a} + \frac{B}{(x-a)^2} \) for factors.
You’ve mastered advanced integration! Next, we’ll explore special functions. 🎉