Question

Find the indefinite integral. (use c for the constant of integration.) sin xcos5 x dx

Answer 1

`\displaystyle \int {sin(x)cos(5x)} \, dx = (cos(4x))/(8) - (cos(6x))/(12) + C`

General Formulas and Concepts:

Algebra I

• Terms/Coefficients
• Expanding/Factoring

Pre-Calculus

Trigonometric Identities

• Product-to-Sum Formula:
  `\displaystyle sin(x)cos(y) = (sin(y + x) - sin(y - x))/(2)`

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

Integration Property [Addition/Subtraction]:
`\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx`

U-Substitution

Explanation:

Step 1: Define

Identify

`\displaystyle \int {sin(x)cos(5x)} \, dx`

Step 2: Integrate Pt. 1

1. [Integrand] Rewrite [Product-to-Sum Formula]:
   `\displaystyle \int {sin(x)cos(5x)} \, dx = \int {(sin(6x) - sin(4x))/(6)} \, dx`
2. [Integrand] Rewrite:
   `\displaystyle \int {sin(x)cos(5x)} \, dx = \int {\Big( (sin(6x))/(2) - (sin(4x))/(2) \Big)} \, dx`
3. [Integral] Rewrite [Integration Property - Addition/Subtraction]:
   `\displaystyle \int {sin(x)cos(5x)} \, dx = \int {(sin(6x))/(2)} \, dx - \int {(sin(4x))/(2)} \, dx`
4. [Integrals] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int {sin(x)cos(5x)} \, dx = (1)/(2)\int {sin(6x)} \, dx - (1)/(2)\int {sin(4x)} \, dx`
5. Factor:
   `\displaystyle \int {sin(x)cos(5x)} \, dx = (1)/(2) \bigg[ \int {sin(6x)} \, dx - \int {sin(4x)} \, dx \bigg]`

Step 3: integrate Pt. 2

Identify variables for u-substitution.

Integral 1:

1. Set u:
   `\displaystyle u = 6x`
2. [u] Differentiate [Basic Power Rule, Multiplied Constant]:
   `\displaystyle du = 6 \ dx`

Integral 2:

1. Set z:
   `\displaystyle z = 4x`
2. [z] Differentiate [Basic Power Rule, Multiplied Constant]:
   `\displaystyle dz = 4 \ dx`

Step 4: Integrate Pt. 3

1. [Integrals] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int {sin(x)cos(5x)} \, dx = (1)/(2) \bigg[ (1)/(6)\int {6sin(6x)} \, dx - (1)/(4)\int {4sin(4x)} \, dx \bigg]`
2. [Integrals] U-Substitution:
   `\displaystyle \int {sin(x)cos(5x)} \, dx = (1)/(2) \bigg[ (1)/(6)\int {sin(u)} \, du - (1)/(4)\int {sin(z)} \, dz \bigg]`
3. [Integrals] Trigonometric Integration:
   `\displaystyle \int {sin(x)cos(5x)} \, dx = (1)/(2) \bigg[ (1)/(6)[-cos(u)] - (1)/(4)[-cos(z)] \bigg] + C`
4. Simplify:
   `\displaystyle \int {sin(x)cos(5x)} \, dx = (1)/(2) \bigg[ (cos(z))/(4) - (cos(u))/(6) \bigg] + C`
5. Expand:
   `\displaystyle \int {sin(x)cos(5x)} \, dx = (cos(z))/(8) - (cos(u))/(12) + C`
6. Back-Substitute:
   `\displaystyle \int {sin(x)cos(5x)} \, dx = (cos(4x))/(8) - (cos(6x))/(12) + C`

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

Unit: Integration