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Answer:

\(\displaystyle \int {e^{2x}cos(3x)} \, dx = \frac{e^{2x}}{13} \bigg[ 3sin(3x) + 2cos(3x) \bigg] + C\)

General Formulas and Concepts:

Algebra I

  • Terms/Coefficients
  • Factoring

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Rule [Chain Rule]:                                                                                 \(\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)\)

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

Integration by Parts:                                                                                               \(\displaystyle \int {u} \, dv = uv - \int {v} \, du\)

  • [IBP] LIPET: Logs, inverses, Polynomials, Exponentials, Trig

Step-by-step explanation:

Step 1: Define

Identify

\(\displaystyle \int {e^{2x}cos(3x)} \, dx\)

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

  1. Set u:                                                                                                             \(\displaystyle u = e^{2x}\)
  2. [u] Differentiate [Exponential Differentiation, Chain Rule]:                         \(\displaystyle du = 2e^{2x} \ dx\)
  3. Set dv:                                                                                                           \(\displaystyle dv = cos(3x) \ dx\)
  4. [dv] Trigonometric Integration [U-Substitution]:                                         \(\displaystyle v = \frac{sin(3x)}{3}\)

Step 3: Integrate Pt. 2

  1. [Integral] Integration by Parts:                                                                       \(\displaystyle \int {e^{2x}cos(3x)} \, dx = \frac{e^{2x}sin(3x)}{3} - \int {\frac{2e^{2x}sin(3x)}{3}} \, dx\)
  2. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 \(\displaystyle \int {e^{2x}cos(3x)} \, dx = \frac{e^{2x}sin(3x)}{3} - \frac{2}{3}\int {e^{2x}sin(3x)} \, dx\)

Step 4: Integrate Pt. 3

Identify variables for integration by parts using LIPET (again).

  1. Set u:                                                                                                             \(\displaystyle u = e^{2x}\)
  2. [u] Differentiate [Exponential Differentiation, Chain Rule]:                         \(\displaystyle du = 2e^{2x} \ dx\)
  3. Set dv:                                                                                                           \(\displaystyle dv = sin(3x) \ dx\)
  4. [dv] Trigonometric Integration [U-Substitution]:                                         \(\displaystyle v = \frac{-cos(3x)}{3}\)

Step 5: Integrate Pt. 4

  1. [Integral] Integration by Parts:                                                                       \(\displaystyle \int {e^{2x}cos(3x)} \, dx = \frac{e^{2x}sin(3x)}{3} - \frac{2}{3} \bigg[ \frac{-e^{2x}cos(3x)}{3} - \int {\frac{-2e^{2x}cos(3x)}{3}} \, dx \bigg]\)
  2. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 \(\displaystyle \int {e^{2x}cos(3x)} \, dx = \frac{e^{2x}sin(3x)}{3} - \frac{2}{3} \bigg[ \frac{-e^{2x}cos(3x)}{3} + \frac{2}{3}\int {e^{2x}cos(3x)} \, dx \bigg]\)
  3. Simplify:                                                                                                         \(\displaystyle \int {e^{2x}cos(3x)} \, dx = \frac{e^{2x}sin(3x)}{3} + \frac{2e^{2x}cos(3x)}{9} - \frac{4}{9}\int {e^{2x}cos(3x)} \, dx\)
  4. Rewrite:                                                                                                         \(\displaystyle \frac{13}{9}\int {e^{2x}cos(3x)} \, dx = \frac{e^{2x}sin(3x)}{3} + \frac{2e^{2x}cos(3x)}{9} + C\)
  5. Isolate:                                                                                                           \(\displaystyle \int {e^{2x}cos(3x)} \, dx = \frac{9}{13} \bigg[ \frac{e^{2x}sin(3x)}{3} + \frac{2e^{2x}cos(3x)}{9} \bigg] + C\)
  6. Simplify:                                                                                                         \(\displaystyle \int {e^{2x}cos(3x)} \, dx = \frac{3e^{2x}sin(3x)}{13} + \frac{2e^{2x}cos(3x)}{13} + C\)
  7. Factor:                                                                                                           \(\displaystyle \int {e^{2x}cos(3x)} \, dx = \frac{e^{2x}}{13} \bigg[ 3sin(3x)} + 2cos(3x) \bigg] + C\)

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

Unit: Integration

Book: College Calculus 10e

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