Verified Answer :

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Answer:

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

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

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

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Integration

  • Integrals
  • [Indefinite Integrals] Integration Constant C

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

U-Substitution

Step-by-step explanation:

Step 1: Define

Identify

\(\displaystyle \int {6sin(3t)} \, dt\)

Step 2: Integrate Pt. 1

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

Step 3: Integrate Pt. 2

Identify variables for u-substitution.

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

Step 4: integrate Pt. 3

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

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

Unit:  Integration