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emmainna [20.7K]
2 years ago
11

Evaluate the following definite integral​

Mathematics
1 answer:
mihalych1998 [28]2 years ago
8 0

Answer:

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

General Formulas and Concepts:

<u>Symbols</u>

  • e (Euler's number) ≈ 2.71828

<u>Algebra I</u>

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

<u>Calculus</u>

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]:                                                               \displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C

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

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

U-Substitution

  • U-Solve

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

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

Step-by-step explanation:

<u>Step 1: Define</u>

<em>Identify</em>

\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx

<u>Step 2: Integrate Pt. 1</u>

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

<u>Step 3: Integrate Pt. 2</u>

<em>Identify variables for u-solve.</em>

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

<u>Step 4: Integrate Pt. 3</u>

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

<u>Step 5: integrate Pt. 4</u>

<em>Identify variables for integration by parts using LIPET.</em>

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

<u>Step 6: Integrate Pt. 5</u>

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

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

Unit: Integration

Book: College Calculus 10e

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