Mathematics, 02.11.2019 17:31 josie311251

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

Answers: 3

Show answers

Answers

Answer from: norahramahi123

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

General Formulas and Concepts:

Algebra I

Terms/CoefficientsFactoring

Calculus

Differentiation

DerivativesDerivative Notation

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

Integration

IntegralsIndefinite IntegralsIntegration 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.

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

Step 3: Integrate Pt. 2

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

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

Step 5: Integrate Pt. 4

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

Answer from: Quest

The answer would be 484

Answer from: Quest

600,000,000,000, or 6e14

step-by-step explanation:

Another question on Mathematics

Mathematics, 21.06.2019 15:10

What is the area of triangle abc if a = 8, b = 11, and c = 15?

Answers: 1

Answer

Mathematics, 21.06.2019 18:30

Factor k2 - 17k + 16. a.(k - 2)(k - 8) b.(k - 1)(k + 16) c.(k - 1)(k - 16)

Answers: 1

Answer

Mathematics, 21.06.2019 19:00

Which sentence uses an objective tone? we will plant the best garden of any school in the area. the new campus garden will surely be envied by other schools. a garden would provide numerous benefits to the students. you will probably agree that gardens have many benefits.

Answers: 3

Answer

Mathematics, 21.06.2019 19:00

65% of students in your school participate in at least one after school activity. if there are 980 students in you school, how many do not participate in an after school activity?

Answers: 1

Answer

You know the right answer?

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

Questions

Mathematics, 05.11.2020 01:00

What do I need to know to ace my test on basic constructions for geometry...

Mathematics, 05.11.2020 01:00

Choose the representations or descriptions that model the relation provided in each item. For each relation, select all situations that apply. x -2 0...