Unit 6 · Core unit

Techniques of Integration and Differential Equations

The honors extension. Recognising which technique an integrand calls for is the skill; differential equations then turn integration back into modeling.

What a strong answer looks like

A strong answer says why a technique was chosen, and for a differential equation states the general solution before applying the initial condition.

Practice this unit Course overview

Chapter introduction

Understand the idea before memorizing the rule

The honors extension. Recognising which technique an integrand calls for is the skill; differential equations then turn integration back into modeling.

In Calculus Honors, success means moving among words, symbols, tables, and graphs while preserving the meaning of each quantity. Use the lessons below as connected notes: explain the concept, carry out the procedure, test the result, and interpret it in context.

Detailed study notes

Core lessons, reasoning, and common errors

Lesson 1

Choosing a Technique

Concept explanation

Substitution handles an inner function with its derivative; parts handles a product where one factor simplifies when differentiated; partial fractions handles a factorable rational.

How to apply it

Classify the integrand before integrating: is it a composition, a product, a rational function, or a form needing a trigonometric substitution?

Common mistake

Reaching for parts when substitution would work, and looping without simplifying.

Guided study move

Classify twelve integrands by technique without evaluating any of them.

Lesson 2

Improper Integrals and Approximation

Concept explanation

An improper integral is defined by a limit, and an infinite region can still have finite area. Numerical rules approximate when no antiderivative is available.

How to apply it

Replace the infinite limit with a variable, evaluate, then take the limit and say whether it converges.

Common mistake

Evaluating an improper integral as though the limit were an ordinary endpoint.

Guided study move

Compare two improper integrals where one converges and the other does not.

Lesson 3

Separable Differential Equations

Concept explanation

A differential equation describes a relationship between a function and its rate. Separating gathers each variable with its own differential before integrating.

How to apply it

Separate, integrate both sides, include the constant, then apply the initial condition to pin it down.

Common mistake

Applying the initial condition before integrating, or losing the constant entirely.

Guided study move

Solve a separable equation with an initial condition, then check it satisfies the original equation.

Formula and visual reference

Connect the symbols to the picture

Chapter synthesis

Ideas to connect

Classifying before integrating
Defining improper integrals as limits
Applying the initial condition last

Modeling lab

Transfer the chapter to a new setting

A population grows at a rate proportional to its size with a known starting value. Set up the differential equation, solve it, and interpret the constant.

Mastery checklist

• Classify an integrand by technique.
• Integrate by parts.
• Decide whether an improper integral converges.
• Solve a separable equation with an initial condition.

Check yourself

• How do you recognise that parts, rather than substitution, is needed?
• Why must the constant of integration appear before the initial condition is used?

Modeling drill

A population grows at a rate proportional to its size with a known starting value. Set up the differential equation, solve it, and interpret the constant.

Integration by partsPartial fractionsTrigonometric substitutionImproper integralConvergenceSeparable equationSlope fieldInitial condition

Unit 5: Applications of Integration