Basic Integration Rules and Antiderivatives

Common Indefinite Integrals

Indefinite integrals, or antiderivatives, are fundamental in calculus for finding functions whose derivatives yield the given integrand. Below are key formulas:

• Power Rule: , for

• Reciprocal Rule:

• Exponential Rule:

• Trigonometric Rules:

Linearity of Integration: Integration is linear, meaning:

Basic Functions and Their Properties

Graphs and Limiting Behavior

Understanding the behavior and graphs of basic functions is essential for calculus. Below are some key functions:

• Cosine Function:

  • Period:

  • Range:

  • Key points: , ,

• Sine Function:

  • Period:

  • Range:

  • Key points: , , , ,

• Exponential Growth:

• Exponential Decay:

• Logarithmic Growth:

Integration Techniques

Substitution (u-Substitution)

Substitution is used to simplify integrals by changing variables. The method is especially useful when the integrand contains a function and its derivative.

• Let , then

• Rewrite the integral in terms of and

• After integrating, substitute back to the original variable

Example: Let , , so

Integration by Parts

Integration by parts is based on the product rule for differentiation and is used for integrals involving products of functions.

• Formula:

• Choose and such that and are easily computable

Example: Let , ,

Partial Fraction Decomposition

Partial fractions are used to break down rational functions into simpler fractions that can be integrated individually.

• If is a proper rational expression, decompose into a sum of simpler fractions.

• Each factor in generates a term in the decomposition.

Example:

Applications of Integration

Definite Integrals and Area

Definite integrals are used to compute the area under a curve between two points, as well as other physical quantities such as total energy consumption or mass.

• gives the net area under from to

• Units of the definite integral are the product of the units of and

Example: If is a rate of energy consumption in units of kW, then gives total energy consumed in kWh over 24 hours.

Physical Applications: Mass by Integration

Integration can be used to find the mass of a planet with variable density:

• Density function:

• Mass:

Advanced Integration Techniques

Trigonometric Integrals

Integrals involving trigonometric functions often require identities or substitutions.

• Half-angle identities:

• Substitution for integrals like or

Integration by Substitution and Parts: Strategy Table

Summary Table: Common Antiderivatives

Key Concepts and Strategies

• Antiderivatives are unique up to a constant: If is an antiderivative of , then is also an antiderivative for any constant .

• Choosing the right technique: Use substitution when the integrand contains a function and its derivative; use integration by parts for products of functions; use partial fractions for rational functions.

• Physical interpretation: Definite integrals often represent total quantities, such as area, mass, or energy, depending on the context.

Example Application: To find the total energy consumed over a day, integrate the rate function over the interval .

Additional info: These notes summarize key integration techniques, basic function properties, and applications as covered in a standard Calculus II course. For more advanced topics, such as improper integrals or differential equations, consult further chapters.