Understanding the Greatest Integer Function and Integration

The video tutorial explains the greatest integer function and defines a real-valued function over the interval [-10, 10]. It visualizes the function's behavior and explores the periodic nature of the cosine function. The tutorial simplifies the integral of the product of these functions using symmetry and evaluates it using integration by parts, ultimately finding the result to be 4.

What does the greatest integer function represent for a real number x?

• The average of integers around x

• The largest integer less than or equal to x

• The absolute value of x

• The smallest integer greater than or equal to x

How is the function f(x) defined when the greatest integer of x is odd?

• f(x) = x + the greatest integer of x

• f(x) = x - the greatest integer of x

• f(x) = 1 + the greatest integer of x - x

• f(x) = x * the greatest integer of x

What is the behavior of the function f(x) between the intervals 0 to 1?

• f(x) = x

• f(x) = x - 1

• f(x) = 1 - x

• f(x) = x + 1

What is the periodicity of the cosine function in the context of this problem?

• It is not periodic

• It has a period of 2

• It has a period of pi

• It has a period of 1

How is the integral from -10 to 10 simplified using symmetry?

• By evaluating from -10 to 0 and multiplying by 2

• By evaluating from 0 to 10 and dividing by 2

• By evaluating only from 0 to 1 and multiplying by 20

• By evaluating from -5 to 5 and multiplying by 4

What is the significance of the symmetry in the function f(x) and cosine(pi*x)?

• It simplifies the function to a constant

• It changes the period of the function

• It makes the function non-integrable

• It allows the integral to be evaluated over a smaller interval

What is the integral of cosine(pi*x) from 0 to 1?

• 1/pi

• 1

• pi

• 0

Which method is used to evaluate the integral of x*cos(pi*x)?

• Partial fractions

• Trigonometric identities

• Substitution

• Integration by parts

What is the antiderivative of sine(pi*x)?

• pi * sine(pi*x)

• -1/pi * cosine(pi*x)

• pi * cosine(pi*x)

• 1/pi * sine(pi*x)

What is the final result of the integral evaluation?

• 2

• 4

• pi

• 0