Answer:
[tex]\displaystyle \int {(3x + 4)^2} \, dx = \frac{(3x + 4)^3}{9} + C[/tex]
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Integration
- Integrals
- Indefinite Integrals
- Integration Constant C
Integration Rule [Reverse Power Rule]: [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Property [Multiplied Constant]: [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
U-Substitution
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle \int {(3x + 4)^2} \, dx[/tex]
Step 2: Integrate Pt. 1
Identify variables for u-substitution.
- Set u: [tex]\displaystyle u = 3x + 4[/tex]
- [u] Differentiate [Basic Power Rule]: [tex]\displaystyle du = 3 \ dx[/tex]
Step 3: Integrate Pt. 2
- [Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle \int {(3x + 4)^2} \, dx = \frac{1}{3}\int {3(3x + 4)^2} \, dx[/tex]
- [Integral] U-Substitution: [tex]\displaystyle \int {(3x + 4)^2} \, dx = \frac{1}{3}\int {u^2} \, du[/tex]
- [Integral] Reverse Power Rule: [tex]\displaystyle \int {(3x + 4)^2} \, dx = \frac{1}{3} \bigg( \frac{u^3}{3} \bigg) + C[/tex]
- Simplify: [tex]\displaystyle \int {(3x + 4)^2} \, dx = \frac{u^3}{9} + C[/tex]
- Back-Substitute: [tex]\displaystyle \int {(3x + 4)^2} \, dx = \frac{(3x + 4)^3}{9} + C[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
Book: College Calculus 10e
