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evaluate the following definite integral​

Evaluate The Following Definite Integral class=

Sagot :

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Answer:

[tex]\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \frac{e}{3}[/tex]

General Formulas and Concepts:

Symbols

  • e (Euler's number) ≈ 2.71828

Algebra I

  • Exponential Rule [Multiplying]:                                                                     [tex]\displaystyle b^m \cdot b^n = b^{m + n}[/tex]

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Integration

  • Integrals
  • Definite Integrals
  • Integration Constant C

Integration Rule [Reverse Power Rule]:                                                               [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]

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

  • U-Solve

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 {x^5e^{x^3 + 1}} \, dx[/tex]

Step 2: Integrate Pt. 1

  1. [Integrand] Rewrite [Exponential Rule - Multiplying]:                                 [tex]\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \int\limits^1_0 {x^5e^{x^3}e} \, dx[/tex]
  2. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 [tex]\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = e\int\limits^1_0 {x^5e^{x^3}} \, dx[/tex]

Step 3: Integrate Pt. 2

Identify variables for u-solve.

  1. Set u:                                                                                                             [tex]\displaystyle u = x^3[/tex]
  2. [u] Differentiate [Basic Power Rule]:                                                             [tex]\displaystyle du = 3x^2 \ dx[/tex]
  3. [u] Rewrite:                                                                                                     [tex]\displaystyle x = \sqrt[3]{u}[/tex]
  4. [du] Rewrite:                                                                                                   [tex]\displaystyle dx = \frac{1}{3x^2} \ du[/tex]

Step 4: Integrate Pt. 3

  1. [Integral] U-Solve:                                                                                         [tex]\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = e\int\limits^1_0 {x^5e^{(\sqrt[3]{u})^3}\frac{1}{3x^2}} \, du[/tex]
  2. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 [tex]\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \frac{e}{3}\int\limits^1_0 {x^5e^{(\sqrt[3]{u})^3}\frac{1}{x^2}} \, du[/tex]
  3. [Integral] Simplify:                                                                                         [tex]\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \frac{e}{3}\int\limits^1_0 {x^3e^u} \, du[/tex]
  4. [Integrand] U-Solve:                                                                                      [tex]\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \frac{e}{3}\int\limits^1_0 {ue^u} \, du[/tex]

Step 5: integrate Pt. 4

Identify variables for integration by parts using LIPET.

  1. Set u:                                                                                                             [tex]\displaystyle u = u[/tex]
  2. [u] Differentiate [Basic Power Rule]:                                                             [tex]\displaystyle du = du[/tex]
  3. Set dv:                                                                                                           [tex]\displaystyle dv = e^u \ du[/tex]
  4. [dv] Exponential Integration:                                                                         [tex]\displaystyle v = e^u[/tex]

Step 6: Integrate Pt. 5

  1. [Integral] Integration by Parts:                                                                        [tex]\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \frac{e}{3} \bigg[ ue^u \bigg| \limits^1_0 - \int\limits^1_0 {e^u} \, du \bigg][/tex]
  2. [Integral] Exponential Integration:                                                               [tex]\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \frac{e}{3} \bigg[ ue^u \bigg| \limits^1_0 - e^u \bigg| \limits^1_0 \bigg][/tex]
  3. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:           [tex]\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \frac{e}{3}[ e - e ][/tex]
  4. Simplify:                                                                                                         [tex]\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \frac{e}{3}[/tex]

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

Unit: Integration

Book: College Calculus 10e

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