Integrating Trigonometric Functions

The video tutorial explains how to find the integral of sine x cosine x using three different methods: u-substitution with sine, u-substitution with cosine, and the double angle formula. Each method is detailed with step-by-step instructions, and the results are compared. The tutorial also explores different forms of the double angle formula for cosine 2x, demonstrating how they can be used to convert between the different integral forms.

What is the first step in the u-substitution method when u is set to sine x?

• Replace u with cosine x

• Replace du with sine x dx

• Replace sine x with u

• Replace dx with du

What is the final expression for the integral using the first method?

• One half sine squared x plus c

• Negative one half cosine squared x plus c

• One fourth sine squared x plus c

• Negative one fourth cosine two x plus c

In the second method, what is the derivative of u when u is set to cosine x?

• Cosine x dx

• Negative sine x dx

• Negative cosine x dx

• Sine x dx

What is the final expression for the integral using the second method?

• Negative one half cosine squared x plus c

• One half sine squared x plus c

• One fourth sine squared x plus c

• Negative one fourth cosine two x plus c

What formula is used in the third method to find the integral?

• Double angle formula for cosine 2x

• Double angle formula for sine 2x

• Power rule for integration

• Product rule for differentiation

What is the antiderivative of sine 2x in the third method?

• Negative sine 2x divided by 2

• Negative cosine 2x divided by 2

• Cosine 2x divided by 2

• Sine 2x divided by 2

Which of the following is a form of the double angle formula for cosine 2x?

• 1 + 2 cosine squared x

• 2 cosine squared x + 1

• 2 sine squared x - 1

• 1 - 2 sine squared x

How can the expression negative one fourth cosine 2x be rewritten using the double angle formula?

• One half cosine squared x

• Negative one half cosine squared x

• One half sine squared x

• Negative one half sine squared x

What is the relationship between the different forms of the integral of sine x cosine x?

• They are equivalent only when using the product rule

• They are equivalent and can be converted using double angle formulas

• They are unrelated and cannot be converted

• They are only equivalent when using the power rule

What is the significance of the constant c in the integral expressions?

• It is only used in the first method

• It represents a specific value

• It is a generic constant that can be adjusted

• It is always zero