Mathematics, 02.11.2019 17:31, josie311251

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

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answered: 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

answered: Guest

d is the correct answer

step-by-step explanation:

x= 8 because 3x = 3 * 8, and 3*8 = 24

answered: Guest

no association

step-by-step explanation:

if you have eyes and you can use them efficiently you can see that it just looks like random dots like do a line of best fit and like they're so spread out it's such a weak association you might as well consider it nonexistent