Question

Evaluate the following definite integral​

Answer 1

`\displaystyle \int\limits^1_0 {x^5e^(x^3 + 1)} \, dx = (e)/(3)`

General Formulas and Concepts:

Symbols

• e (Euler's number) ≈ 2.71828

Algebra I

• Exponential Rule [Multiplying]:
  `\displaystyle b^m \cdot b^n = b^(m + n)`

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]:
`\displaystyle \int {x^n} \, dx = (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

Explanation:

Step 1: Define

Identify

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

Step 2: Integrate Pt. 1

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`

Step 3: Integrate Pt. 2

Identify variables for u-solve.

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

Step 4: Integrate Pt. 3

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

Step 5: integrate Pt. 4

Identify variables for integration by parts using LIPET.

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

Step 6: Integrate Pt. 5

1. [Integral] Integration by Parts:
   `\displaystyle \int\limits^1_0 {x^5e^(x^3 + 1)} \, dx = (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 = (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 = (e)/(3)[ e - e ]`
4. Simplify:
   `\displaystyle \int\limits^1_0 {x^5e^(x^3 + 1)} \, dx = (e)/(3)`

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

Unit: Integration

Book: College Calculus 10e