Answered

Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

What is the indefinite integral of 6sin(3t) dt?

Sagot :

Space

Answer:

[tex]\displaystyle \int {6sin(3t)} \, dt = -2cos(3t) + C[/tex]

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Property [Multiplied Constant]:                                                           [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Integration

  • Integrals
  • [Indefinite Integrals] Integration Constant C

Integration Property [Multiplied Constant]:                                                         [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

U-Substitution

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle \int {6sin(3t)} \, dt[/tex]

Step 2: Integrate Pt. 1

  1. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 [tex]\displaystyle \int {6sin(3t)} \, dt = 6\int {sin(3t)} \, dt[/tex]

Step 3: Integrate Pt. 2

Identify variables for u-substitution.

  1. Set u:                                                                                                             [tex]\displaystyle u = 3t[/tex]
  2. [u] Differentiate [Basic Power Rule, Multiplied Constant]:                         [tex]\displaystyle du = 3 \ dt[/tex]

Step 4: integrate Pt. 3

  1. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 [tex]\displaystyle \int {6sin(3t)} \, dt = 2\int {3sin(3t)} \, dt[/tex]
  2. [Integral] U-Substitution:                                                                               [tex]\displaystyle \int {6sin(3t)} \, dt = 2\int {sin(u)} \, du[/tex]
  3. [Integral] Trigonometric Integration:                                                             [tex]\displaystyle \int {6sin(3t)} \, dt = -2cos(u) + C[/tex]
  4. Back-Substitute:                                                                                             [tex]\displaystyle \int {6sin(3t)} \, dt = -2cos(3t) + C[/tex]

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit:  Integration

Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.