Question

Integrate the following (3x+4)^2​

Answer 1

`\displaystyle \int {(3x + 4)^2} \, dx = ((3x + 4)^3)/(9) + C`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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
• Indefinite Integrals
• Integration Constant C

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

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

U-Substitution

Explanation:

Step 1: Define

Identify

`\displaystyle \int {(3x + 4)^2} \, dx`

Step 2: Integrate Pt. 1

Identify variables for u-substitution.

1. Set u:
   `\displaystyle u = 3x + 4`
2. [u] Differentiate [Basic Power Rule]:
   `\displaystyle du = 3 \ dx`

Step 3: Integrate Pt. 2

1. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int {(3x + 4)^2} \, dx = (1)/(3)\int {3(3x + 4)^2} \, dx`
2. [Integral] U-Substitution:
   `\displaystyle \int {(3x + 4)^2} \, dx = (1)/(3)\int {u^2} \, du`
3. [Integral] Reverse Power Rule:
   `\displaystyle \int {(3x + 4)^2} \, dx = (1)/(3) \bigg( (u^3)/(3) \bigg) + C`
4. Simplify:
   `\displaystyle \int {(3x + 4)^2} \, dx = (u^3)/(9) + C`
5. Back-Substitute:
   `\displaystyle \int {(3x + 4)^2} \, dx = ((3x + 4)^3)/(9) + C`

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

Unit: Integration

Book: College Calculus 10e