contestada

Determine whether the statement is true or false, and explain why.
Integration by parts should be used to evaluate ∫^1_0 xe^10x dx.

Respuesta :

Answer:

True

Step-by-step explanation:

The integral

[tex]\int\limits^1_0 {xe^{10x}} \, dx[/tex]

in the integral we have a product of a monomial: [tex]x[/tex]

and an exponential function: [tex]e^{10x}[/tex]

In general case, when we have a combination of these two things you can use the integration by parts, where [tex]x[/tex] will be [tex]u[/tex] and [tex]e^{10x}[/tex] will be [tex]dv[/tex].

The statement is true

Space

Answer:

True

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Integration

  • Integrals

Integration Rule [Fundamental Theorem of Calculus 1]:                                     [tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]

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\limits^1_0 {xe^{10x}} \, 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] Basic Power Rule:                                                                                     [tex]\displaystyle du = dx[/tex]
  3. Set dv:                                                                                                           [tex]\displaystyle dv = e^{10x} \ dx[/tex]
  4. [dv] Exponential Integration [U-Substitution]:                                             [tex]\displaystyle v = \frac{e^{10x}}{10}[/tex]

Step 3: integrate Pt. 2

  1. [Integral] Integration by Parts:                                                                       [tex]\displaystyle \int\limits^1_0 {xe^{10x}} \, dx = \frac{xe^{10x}}{10} \bigg| \limits^1_0 - \int\limits^1_0 {\frac{e^{10x}}{10}} \, dx[/tex]
  2. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 [tex]\displaystyle \int\limits^1_0 {xe^{10x}} \, dx = \frac{xe^{10x}}{10} \bigg| \limits^1_0 - \frac{1}{10} \int\limits^1_0 {e^{10x}} \, dx[/tex]
  3. [Integral] Exponential Integration:                                                               [tex]\displaystyle \int\limits^1_0 {xe^{10x}} \, dx = \frac{xe^{10x}}{10} \bigg| \limits^1_0 - \frac{1}{10} \bigg( \frac{e^{10x}}{10} \bigg) \bigg| \limits^1_0[/tex]
  4. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:          [tex]\displaystyle \int\limits^1_0 {xe^{10x}} \, dx = \frac{9e^{10}}{100} + \frac{1}{10}[/tex]
  5. Simplify:                                                                                                         [tex]\displaystyle \int\limits^1_0 {xe^{10x}} \, dx = \frac{9e^{10} + 1}{100}[/tex]

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

Unit: Integration