Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
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
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.