Indefinite Integral of Trigonometric Functions

The video tutorial demonstrates how to solve the indefinite integral of sine squared x times cosine cubed x using algebraic manipulation and u-substitution. The instructor explains the initial challenges and how to overcome them by separating terms and applying the Pythagorean Identity. The process involves simplifying the expression and using u-substitution to find the anti-derivative, leading to the final solution.

What is the initial problem presented in the video?

• Solving a trigonometric equation

• Finding the indefinite integral of sine squared x cosine to the third x

• Finding the limit of sine squared x cosine to the third x

• Finding the derivative of sine squared x cosine to the third x

Why can't u-substitution be directly applied to the given integral?

• Because neither sine nor cosine has an exponent of one

• Because both sine and cosine have even exponents

• Because the integral is already in its simplest form

• Because both sine and cosine have odd exponents

What is the strategy used to handle the odd exponent in the integral?

• Use partial fraction decomposition

• Convert everything to sine

• Use integration by parts

• Separate one cosine and use the Pythagorean Identity

Tags

CCSS.HSF.TF.C.9

Which identity is used to simplify the expression in the integral?

• Double Angle Identity

• Angle Addition Identity

• Sum-to-Product Identity

• Pythagorean Identity

What substitution is made to apply u-substitution?

• u = secant of x

• u = tangent of x

• u = sine of x

• u = cosine of x

What does the integral become after applying u-substitution?

• u squared plus u to the fourth

• u squared minus u to the fourth

• u cubed plus u to the fifth

• u cubed minus u to the fifth

What is the antiderivative of u squared?

• u to the third over three

• u to the fourth over four

• u to the fifth over five

• u squared over two

What is the antiderivative of u to the fourth?

• u to the sixth over six

• u to the fifth over five

• u to the fourth over four

• u to the third over three

What is the final expression for the integral after reverse substitution?

• secant of x to the third over three minus secant of x to the fifth over five

• sine of x to the third over three minus sine of x to the fifth over five

• cosine of x to the third over three minus cosine of x to the fifth over five

• tangent of x to the third over three minus tangent of x to the fifth over five

What constant is added to the final expression of the integral?

• c

• d

• b

• a