Question

Integrate the following: e^(2x) cos3x dx

Answer 1

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

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 = (sin(3x))/(3)`

Step 3: Integrate Pt. 2

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

Step 5: Integrate Pt. 4

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