contestada

Evaluate the integral using integration by parts with the indicated choices of u and dv. (Use C for the constant of integration.) xe9x dx; u

Respuesta :

Answer:

The answer is "[tex]\bold{\frac{ \ e^{9x}}{9} (x-1)+C}\\[/tex]"

Step-by-step explanation:

Taking the integral [tex]\int x^{2x} \ dx[/tex]  

Evaluating the integral by using integration:

[tex]u=x\\\\dv=e^{9x} \ dx[/tex]

Let

[tex]u=x\\\\ dv=e^{9x} dx[/tex]

Then,

[tex]\to \frac{du}{dx}= 1 \ and\ v =\int e^{9X} \ dx \\\\\to du = 1 \ dx \\\\\to v = \frac{e^{9x}}{9}\\\\\text{integrate the above value}:\\\\\int\ u \ dv = uv - \int v du \\\\Now, \\\\\to \int x e^{9X} \ dx = x \frac{e^{9x}}{9} - \int \frac{e^{9x}}{9} dx[/tex]

                     [tex]= \frac{ x \ e^{9x}}{9} - \frac{e^{9x}}{9}\\\\= \frac{ \ e^{9x}}{9} (x-1)+C\\[/tex]

Space

Answer:

[tex]\displaystyle \int {xe^{9x}} \, dx = e^{9x} \bigg( \frac{x}{9} - \frac{1}{81} \bigg) + C[/tex]

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Property [Multiplied Constant]:                                                           [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Integration

  • Integrals
  • [Indefinite Integrals] Integration Constant C

Integration Property [Multiplied Constant]:                                                         [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

U-Substitution

Integration by Parts:                                                                                               [tex]\displaystyle \int {u} \, dv = uv - \int {v} \, du[/tex]

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

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle \int {xe^{9x}} \, dx[/tex]

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

  1. Set u:                                                                                                             [tex]\displaystyle u = x[/tex]
  2. [u] Differentiate:                                                                                             [tex]\displaystyle du = dx[/tex]
  3. Set dv:                                                                                                           [tex]\displaystyle dv = e^{9x} \ dx[/tex]
  4. [dv] Integrate [Exponential Integration]:                                                      [tex]\displaystyle v = \frac{e^{9x}}{9}[/tex]

Step 3: Integrate Pt. 2

  1. [Integral] Integration by parts:                                                                       [tex]\displaystyle \int {xe^{9x}} \, dx = \frac{xe^{9x}}{9} - \int {\frac{e^{9x}}{9}} \, du[/tex]
  2. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 [tex]\displaystyle \int {xe^{9x}} \, dx = \frac{xe^{9x}}{9} - \frac{1}{9} \int {e^{9x}} \, du[/tex]

Step 4: Integrate Pt. 3

Identify variables for u-substitution.

  1. Set u:                                                                                                             [tex]\displaystyle u = 9x[/tex]
  2. [u] Differentiate [Derivative Properties, Basic Power Rule]:                       [tex]\displaystyle du = 9 \ dx[/tex]

Step 5: Integrate Pt. 4

  1. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 [tex]\displaystyle \int {xe^{9x}} \, dx = \frac{xe^{9x}}{9} - \frac{1}{81} \int {9e^{9x}} \, du[/tex]
  2. [Integral] U-Substitution:                                                                               [tex]\displaystyle \int {xe^{9x}} \, dx = \frac{xe^{9x}}{9} - \frac{1}{81} \int {e^u} \, du[/tex]
  3. [Integral] Exponential Integration:                                                              [tex]\displaystyle \int {xe^{9x}} \, dx = \frac{xe^{9x}}{9} - \frac{e^u}{81} + C[/tex]
  4. [u] Back-Substitute:                                                                                       [tex]\displaystyle \int {xe^{9x}} \, dx = \frac{xe^{9x}}{9} - \frac{e^{9x}}{81} + C[/tex]
  5. Factor:                                                                                                           [tex]\displaystyle \int {xe^{9x}} \, dx = e^{9x} \bigg( \frac{x}{9} - \frac{1}{81} \bigg) + C[/tex]

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

Unit: Integration

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