Verified Answer :

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Answer:

\(\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{2sin^3(3x)}{9} + C\)

General Formulas and Concepts:

Pre-Calculus

  • Trigonometric Identities

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 Rule [Reverse Power Rule]:                                                               \(\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C\)

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

U-Substitution

Step-by-step explanation:

Step 1: Define

Identify

\(\displaystyle \int {sin(3x)sin(6x)} \, dx\)

Step 2: Integrate Pt. 1

Set variables for u-substitution.

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

Step 3: Integrate Pt. 2

  1. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 \(\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{1}{3} \int {3sin(3x)sin(6x)} \, dx\)
  2. [Integral] U-Substitution:                                                                               \(\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{1}{3} \int {sin(u)sin(2u)} \, du\)
  3. [Integrand] Rewrite [Trigonometric Identities]:                                           \(\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{1}{3} \int {sin(u)[2cos(u)sin(u)]} \, du\)
  4. [Integral] Simplify:                                                                                         \(\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{1}{3} \int {2sin^2(u)cos(u)} \, du\)
  5. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 \(\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{2}{3} \int {sin^2(u)cos(u)} \, du\)

Step 4: Integrate Pt. 3

Set variables for u-substitution #2.

  1. Set z:                                                                                                              \(\displaystyle z = sin(u)\)
  2. [z] Differentiate [Trigonometric Differentiation]:                                         \(\displaystyle dz = -cos(u) \ du\)

Step 5: Integrate Pt. 4

  1. [Integral] U-Substitution:                                                                               \(\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{2}{3} \int {z^2} \, dz\)
  2. [Integral] Reverse Power Rule:                                                                     \(\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{2}{3} \Big( \frac{z^3}{3} \Big) + C\)
  3. Simplify:                                                                                                         \(\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{2z^3}{9} + C\)
  4. [z] Back-Substitute:                                                                                       \(\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{2sin^3(u)}{9} + C\)
  5. [u] Back-Substitute:                                                                                       \(\displaystyle \int {sin(3x)sin(6x)} \, dx = \frac{2sin^3(3x)}{9} + C\)

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

Unit: Integration