Calculus II: Trigonometric Integrals (Level 2 of 7)

This video tutorial covers solving trigonometric integrals where the power of sine is odd. It begins with a review of the previous case where cosine had an odd power and introduces the current case with examples. The video demonstrates using u-substitution and the Pythagorean identity to solve integrals involving powers of sine and cosine. Each example illustrates the process of breaking down the integrand, applying substitution, and integrating using the power rule. The tutorial emphasizes the importance of identifying odd powers and using appropriate identities for simplification.

What is the first step in solving the integral of cosine squared times sine using u-substitution?

• Let u equal cosine of x

• Let u equal sine of x

• Differentiate cosine of x

• Differentiate sine of x

Why is it important to identify a single sine factor in the integrand?

• To differentiate easily

• To make u-substitution possible

• To apply the Pythagorean identity

• To simplify the expression

In the example of sine cubed of 5x, what identity is used to rewrite sine squared?

• Sine squared equals cosine squared

• Sine squared plus cosine squared equals 1

• Sine squared minus cosine squared equals 1

• Sine squared equals 1 minus cosine

What is the purpose of breaking apart an odd power function into an even power and a single factor?

• To apply the chain rule

• To differentiate easily

• To simplify the integration

• To rewrite in terms of cosine

In the third example, what is the first step after identifying the powers of sine and cosine?

• Break apart the odd power

• Use the Pythagorean identity

• Apply the chain rule

• Differentiate the expression

What is the integral of u to the power of 4?

• 1/5 times u to the power of 5

• 1/7 times u to the power of 5

• 1/4 times u to the power of 5

• 1/6 times u to the power of 5

In the final example, what determines the method to solve the integral?

• The highest power of cosine

• The highest power of sine

• The odd power function

• The even power function

What is the final step in solving the integral after using u-substitution?

• Differentiate the result

• Add the constant C

• Substitute back the original function

• Simplify the expression

Why is it necessary to expand the binomial in the final example?

• To apply the chain rule

• To simplify the integration

• To distribute the constant

• To use the Pythagorean identity

What is the integral of u squared?

• 1/5 times u cubed

• 1/4 times u cubed

• 1/3 times u cubed

• 1/2 times u cubed