Question

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

Answer 1

Answer:

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

General Formulas and Concepts:

Algebra I

• Terms/Coefficients
• Expanding/Factoring

Pre-Calculus

Trigonometric Identities

• Product-to-Sum Formula:                                                                             $\displaystyle sin(x)cos(y) = \frac{sin(y + x) - sin(y - x)}{2}$

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 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

Step-by-step explanation:

Step 1: Define

Identify

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

Step 2: Integrate Pt. 1

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

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

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

Unit:  Integration