9.3.4 Choosing Effective Function Decompositions

Question 1

Choosing Effective Function Decompositions

Answer 1

• Experiment with different choices for u when using integration by substitution. A good choice is one whose derivative is expressed elsewhere in the integrand.
• When working with integrands that include trigonometric expressions, it is sometimes necessary to rewrite those expressions using trig identities.

Question 2

note

Answer 2

• When applying integration by substitution to composite
  functions, there may be several choices for u.
• In the case of a rational function, the best choice is often the denominator.
• In this example, du/2 produces the expression in the
  numerator.
• You may want to express trigonometric integrands in terms of sine and cosine before integrating.
• Since the denominator has cosx raised to a power, choose u to be cosx. Then –du produces the expression in the numerator.

Question 3

Which of the following is the best choice for au-substitution for the integral∫2x(x^2−4)^6dx?

Answer 3

u = x ^2 − 4

Question 4

Which of these expressions is the best choice for making a u-substitution for the integral∫sin^32xcos2xdx?

Answer 4

u = sin 2x

Question 5

Which of the following expressions creates a working u-substitution that solves the following integral?
∫x^3sinx^2dx

Answer 5

u = x^ 2

Question 6

What is the best choice for a u-substitution for the integral ∫e^cotx csc^2x dx?

Answer 6

u = cot x