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:
- f(x) = cxⁿ
- 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
- [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.
- Set u: [tex]\displaystyle u = 3t[/tex]
- [u] Differentiate [Basic Power Rule, Multiplied Constant]: [tex]\displaystyle du = 3 \ dt[/tex]
Step 4: integrate Pt. 3
- [Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle \int {6sin(3t)} \, dt = 2\int {3sin(3t)} \, dt[/tex]
- [Integral] U-Substitution: [tex]\displaystyle \int {6sin(3t)} \, dt = 2\int {sin(u)} \, du[/tex]
- [Integral] Trigonometric Integration: [tex]\displaystyle \int {6sin(3t)} \, dt = -2cos(u) + C[/tex]
- Back-Substitute: [tex]\displaystyle \int {6sin(3t)} \, dt = -2cos(3t) + C[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
