Differential Equations and Integration

This page delves deeper into differential equations and introduces the concept of integration.

The page begins by discussing differential equations of the form y' = ay + f, where f is a function.

Definition: The solutions to y' = ay + f are functions of the form y(x) = u(x) + v(x), where u(x) is a particular solution to y' = ay + f, and v(x) is any solution to y' = ay.

A special case is then presented: y' = ay + b, where b is a constant. The particular solutions for this case are constant functions of the form y_p(x) = -b/a.

The page then transitions to the topic of integrals. It provides a geometric interpretation of definite integrals as the area between the curve of a function and the x-axis over a given interval.

Highlight: The fundamental theorem of calculus is introduced: If F(x) = ∫_a^x f(t)dt, then F'(x) = f(x).

Several important properties of integrals are listed, including:

• If f is negative, its integral will also be negative.
• If f is both positive and negative over the interval, the areas should be calculated separately and then added.

The page concludes with a reminder about the trapezoidal rule for approximating integrals.

Example: The area of a trapezoid is given by (small base + large base) × height / 2.

This page provides essential information for students learning about differential equations and integration, bridging the gap between these two fundamental areas of calculus.

Advanced Integration Techniques and Properties

This page focuses on advanced integration techniques and properties, particularly the method of integration by parts.

The page begins with a detailed derivation of the fundamental theorem of calculus, showing how the derivative of an integral function equals the original function.

Highlight: The fundamental theorem of calculus states that if F(x) = ∫_a^x f(t)dt, then F'(x) = f(x).

The page then introduces the concept of integration by parts, a powerful technique for integrating products of functions.

Definition: Integration by parts formula: ∫u(x)v'(x)dx = [u(x)v(x)] - ∫u'(x)v(x)dx

An example is provided to illustrate how to apply this technique:

Example: To integrate x^2 cos(x), we can let u = x^2 and v' = cos(x). This transforms the integral into a more manageable form.

The page also mentions an important property of continuous functions:

Vocabulary: Every continuous function on an interval of R has primitives on that interval, although we may not always be able to find them explicitly (e.g., e^(-x^2)).

This information is crucial for students learning advanced integration techniques and understanding the limitations of analytical methods in calculus.

Properties of Integrals and Trigonometric Identities

This final page provides a comprehensive list of properties of integrals and important trigonometric identities.

The page begins with several key properties of definite integrals:

Highlight:

• ∫_a^b kf(x)dx = k∫_a^b f(x)dx (linearity)
• ∫_a^b [f(x) + g(x)]dx = ∫_a^b f(x)dx + ∫_a^b g(x)dx (additivity)
• ∫_a^c f(x)dx = ∫_a^b f(x)dx + ∫_b^c f(x)dx (additivity over intervals)

Additional properties are listed, including inequalities involving integrals and the concept of average value of a function.

Definition: The average value of a function f on [a,b] is given by (1/(b-a)) ∫_a^b f(x)dx.

The page concludes with a list of important trigonometric identities:

Vocabulary:

• cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
• cos(a-b) = cos(a)cos(b) + sin(a)sin(b)
• sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
• sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
• cos(2a) = cos^2(a) - sin^2(a) = 1 - 2sin^2(a) = 2cos^2(a) - 1
• sin(2a) = 2cos(a)sin(a)

These identities are crucial for solving complex trigonometric problems and are often used in calculus and physics applications.

This page serves as a valuable reference for students working with integrals and trigonometric functions, providing a concise summary of key properties and identities.

Derivatives and Primitives of Trigonometric Functions

This page introduces the fundamental concepts of derivatives and primitives of trigonometric functions, along with a key theorem on primitives.

The page begins with a table showing the derivatives and primitives of sine and cosine functions. This information is crucial for students learning calculus and trigonometry.

Highlight: The derivatives and primitives of sin(x) and cos(x) are presented in a clear, tabular format for easy reference.

A significant theorem is then presented and demonstrated:

Definition: Two primitives of the same function differ by a constant.

The proof of this theorem is provided, showing that if F and G are two primitives of a function f on an interval I, then F(x) - G(x) = C for all x in I, where C is a constant.

The page concludes with an introduction to differential equations of the form y' = ay.

Example: For differential equations of the form y' = ay, the solutions are functions of the form y(x) = Ce^(ax), where C is determined by initial conditions.

This information sets the foundation for more complex topics in differential equations and integration that will be covered in subsequent pages.