Question

Exercise is mixed —some require integration by parts, while others can be integrated by using techniques discussed in the chapter on Integration.

∫^1_0 x^2 dx/2x^3+1

Answer 1

`\displaystyle \int\limits^1_0 {(x^2)/(2x^3 + 1)} \, dx = (\ln 3)/(6)`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Integration

• Integrals

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`

Explanation:

Step 1: Define

Identify

`\displaystyle \int\limits^1_0 {(x^2)/(2x^3 + 1)} \, dx`

Step 2: Integrate Pt. 1

Identify variables for u-substitution.

1. Set u:
   `\displaystyle u = 2x^3 + 1`
2. [u] Basic Power Rule [Derivative Properties]:
   `\displaystyle du = 6x^2 \ dx`
3. [Limits] Swap:
   `\displaystyle \left \{ {{x = 1 ,\ u = 2(1)^3 + 1 = 3} \atop {x = 0 ,\ u = 2(0)^3 + 1 = 1}} \right.`

Step 3: Integrate Pt. 2

1. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int\limits^1_0 {(x^2)/(2x^3 + 1)} \, dx = (1)/(6) \int\limits^1_0 {(6x^2)/(2x^3 + 1)} \, dx`
2. [Integral] U-Substitution:
   `\displaystyle \int\limits^1_0 {(x^2)/(2x^3 + 1)} \, dx = (1)/(6) \int\limits^3_1 {(1)/(u)} \, du`
3. [Integral] Logarithmic Integration:
   `\displaystyle \int\limits^1_0 {(x^2)/(2x^3 + 1)} \, dx = (1)/(6) \ln \big| u \big| \bigg| \limits^3_1`
4. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
   `\displaystyle \int\limits^1_0 {(x^2)/(2x^3 + 1)} \, dx = (\ln 3)/(6)`

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

Unit: Integration