Question

How do I evaluate the indefinite integral
∫sin(3x)⋅sin(6x)dxintsin(3x)*sin(6x)dx ?

Answer 1

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

General Formulas and Concepts:

Pre-Calculus

• Trigonometric Identities

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (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 = (x^(n + 1))/(n + 1) + C`

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

U-Substitution

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 = (1)/(3) \int {3sin(3x)sin(6x)} \, dx`
2. [Integral] U-Substitution:
   `\displaystyle \int {sin(3x)sin(6x)} \, dx = (1)/(3) \int {sin(u)sin(2u)} \, du`
3. [Integrand] Rewrite [Trigonometric Identities]:
   `\displaystyle \int {sin(3x)sin(6x)} \, dx = (1)/(3) \int {sin(u)[2cos(u)sin(u)]} \, du`
4. [Integral] Simplify:
   `\displaystyle \int {sin(3x)sin(6x)} \, dx = (1)/(3) \int {2sin^2(u)cos(u)} \, du`
5. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int {sin(3x)sin(6x)} \, dx = (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 = (2)/(3) \int {z^2} \, dz`
2. [Integral] Reverse Power Rule:
   `\displaystyle \int {sin(3x)sin(6x)} \, dx = (2)/(3) \Big( (z^3)/(3) \Big) + C`
3. Simplify:
   `\displaystyle \int {sin(3x)sin(6x)} \, dx = (2z^3)/(9) + C`
4. [z] Back-Substitute:
   `\displaystyle \int {sin(3x)sin(6x)} \, dx = (2sin^3(u))/(9) + C`
5. [u] Back-Substitute:
   `\displaystyle \int {sin(3x)sin(6x)} \, dx = (2sin^3(3x))/(9) + C`

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

Unit: Integration