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Sagot :
Answer:
[tex]\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{2sin^3(3x)}{9} + C[/tex]
General Formulas and Concepts:
Pre-Calculus
- Trigonometric Identities
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 Rule [Reverse Power Rule]: [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
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 {sin(3x)sin(6x)} \, dx[/tex]
Step 2: Integrate Pt. 1
Set variables for u-substitution.
- Set u: [tex]\displaystyle u = 3x[/tex]
- [u] Differentiate [Basic Power Rule, Multiplied Constant]: [tex]\displaystyle du = 3 \ dx[/tex]
Step 3: Integrate Pt. 2
- [Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{1}{3} \int {3sin(3x)sin(6x)} \, dx[/tex]
- [Integral] U-Substitution: [tex]\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{1}{3} \int {sin(u)sin(2u)} \, du[/tex]
- [Integrand] Rewrite [Trigonometric Identities]: [tex]\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{1}{3} \int {sin(u)[2cos(u)sin(u)]} \, du[/tex]
- [Integral] Simplify: [tex]\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{1}{3} \int {2sin^2(u)cos(u)} \, du[/tex]
- [Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{2}{3} \int {sin^2(u)cos(u)} \, du[/tex]
Step 4: Integrate Pt. 3
Set variables for u-substitution #2.
- Set z: [tex]\displaystyle z = sin(u)[/tex]
- [z] Differentiate [Trigonometric Differentiation]: [tex]\displaystyle dz = -cos(u) \ du[/tex]
Step 5: Integrate Pt. 4
- [Integral] U-Substitution: [tex]\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{2}{3} \int {z^2} \, dz[/tex]
- [Integral] Reverse Power Rule: [tex]\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{2}{3} \Big( \frac{z^3}{3} \Big) + C[/tex]
- Simplify: [tex]\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{2z^3}{9} + C[/tex]
- [z] Back-Substitute: [tex]\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{2sin^3(u)}{9} + C[/tex]
- [u] Back-Substitute: [tex]\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{2sin^3(3x)}{9} + C[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
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