Q:

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

Accepted Solution

A:
Answer:$$\displaystyle \int {sin(x)cos(5x)} \, dx = \frac{cos(4x)}{8} - \frac{cos(6x)}{12} + C$$General Formulas and Concepts:Algebra ITerms/CoefficientsExpanding/FactoringPre-CalculusTrigonometric IdentitiesProduct-to-Sum Formula:                                                                             $$\displaystyle sin(x)cos(y) = \frac{sin(y + x) - sin(y - x)}{2}$$CalculusDifferentiationDerivativesDerivative NotationDerivative Property [Multiplied Constant]:                                                           $$\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$$Basic Power Rule:f(x) = cxⁿf’(x) = c·nxⁿ⁻¹IntegrationIntegrals[Indefinite Integrals] Integration Constant CIntegration 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-SubstitutionStep-by-step explanation:Step 1: DefineIdentify$$\displaystyle \int {sin(x)cos(5x)} \, dx$$Step 2: Integrate Pt. 1[Integrand] Rewrite [Product-to-Sum Formula]:                                           $$\displaystyle \int {sin(x)cos(5x)} \, dx = \int {\frac{sin(6x) - sin(4x)}{6}} \, dx$$[Integrand] Rewrite:                                                                                      $$\displaystyle \int {sin(x)cos(5x)} \, dx = \int {\Big( \frac{sin(6x)}{2} - \frac{sin(4x)}{2} \Big)} \, dx$$[Integral] Rewrite [Integration Property - Addition/Subtraction]:               $$\displaystyle \int {sin(x)cos(5x)} \, dx = \int {\frac{sin(6x)}{2}} \, dx - \int {\frac{sin(4x)}{2}} \, dx$$[Integrals] Rewrite [Integration Property - Multiplied Constant]:               $$\displaystyle \int {sin(x)cos(5x)} \, dx = \frac{1}{2}\int {sin(6x)} \, dx - \frac{1}{2}\int {sin(4x)} \, dx$$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. 2Identify variables for u-substitution.Integral 1:Set u:                                                                                                             $$\displaystyle u = 6x$$[u] Differentiate [Basic Power Rule, Multiplied Constant]:                         $$\displaystyle du = 6 \ dx$$Integral 2:Set z:                                                                                                               $$\displaystyle z = 4x$$[z] Differentiate [Basic Power Rule, Multiplied Constant]:                           $$\displaystyle dz = 4 \ dx$$Step 4: Integrate Pt. 3[Integrals] Rewrite [Integration Property - Multiplied Constant]:               $$\displaystyle \int {sin(x)cos(5x)} \, dx = \frac{1}{2} \bigg[ \frac{1}{6}\int {6sin(6x)} \, dx - \frac{1}{4}\int {4sin(4x)} \, dx \bigg]$$[Integrals] U-Substitution:                                                                            $$\displaystyle \int {sin(x)cos(5x)} \, dx = \frac{1}{2} \bigg[ \frac{1}{6}\int {sin(u)} \, du - \frac{1}{4}\int {sin(z)} \, dz \bigg]$$[Integrals] Trigonometric Integration:                                                           $$\displaystyle \int {sin(x)cos(5x)} \, dx = \frac{1}{2} \bigg[ \frac{1}{6}[-cos(u)] - \frac{1}{4}[-cos(z)] \bigg] + C$$Simplify:                                                                                                         $$\displaystyle \int {sin(x)cos(5x)} \, dx = \frac{1}{2} \bigg[ \frac{cos(z)}{4} - \frac{cos(u)}{6} \bigg] + C$$Expand:                                                                                                         $$\displaystyle \int {sin(x)cos(5x)} \, dx = \frac{cos(z)}{8} - \frac{cos(u)}{12} + C$$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