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Calculus II: Techniques of Integration

Calculus II: Techniques of Integration








Calculus II: Techniques of Integration

> This document covers the fundamental techniques of integration presented in Calculus II. Each module contains theorems, key concepts, and worked examples. Click on an example to view the detailed solution.




Module 1: The Substitution Rule


Theorem: The Substitution Rule. If u=g(x) is a differentiable function whose range is an interval I, and f is continuous on I, then:
∫ f(g(x))g'(x) dx = ∫ f(u) du


Key Concepts:


This technique reverses the Chain Rule for derivatives.

Identify an "inner function" g(x) to define as u.

The derivative of u (i.e., g'(x)dx) must be present in the integral, possibly off by a constant factor.

For definite integrals, convert the limits of integration from x-values to u-values.



Example 1.1: Evaluate ∫ x2√(x3 + 1) dx



1. DEFINE u: Let u be the inner function.
u = x³ + 1

2. FIND du: Differentiate u with respect to x.
du = 3x² dx

3. REWRITE INTEGRAL: Adjust for the constant factor.
(1/3) ∫ √(x³ + 1) * (3x² dx)

4. SUBSTITUTE: Replace all terms with u and du.
(1/3) ∫ √u du = (1/3) ∫ u¹⁄² du

5. INTEGRATE: Apply the power rule for integration.
(1/3) * [ (2/3)u³⁄² ] + C = (2/9)u³⁄² + C

6. BACK-SUBSTITUTE: Convert the result back to x.
(2/9)(x³ + 1)³⁄² + C








Module 2: Integration by Parts


Theorem: Integration by Parts. Derived from the Product Rule for derivatives.
∫ u dv = uv - ∫ v du


Key Concepts:


Used for integrating a product of two functions.

Choose u using the **L.I.P.E.T.** priority list:
**L**ogarithmic, **I**nverse Trig, **P**olynomial, **E**xponential, **T**rigonometric.

The remaining part of the integrand becomes dv.

This process may need to be applied multiple times.



Example 2.1: Evaluate ∫ ln(x) dx



1. CHOOSE u AND dv: Following LIPET, Logarithmic is chosen for u.
u = ln(x) dv = dx
du = (1/x) dx v = x

2. APPLY FORMULA: uv - ∫ v du
x*ln(x) - ∫ x * (1/x) dx

3. SIMPLIFY: The new integral is simpler.
x*ln(x) - ∫ 1 dx

4.…