Question

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

∫(2x-1) ln (3x) dx.

Answer 1

`\displaystyle \int {(2x - 1) \ln(3x)} \, dx = (x^2 - x) \ln(3x) - \bigg( (x^2)/(2) - x \bigg) + C`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Chain Rule]:
`\displaystyle (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`

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`

Integration Property [Addition/Subtraction]:
`\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx`

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 {(2x - 1) \ln(3x)} \, dx`

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

1. Set u:
   `\displaystyle u = \ln(3x)`
2. [u] Logarithmic Differentiation [Derivative Rule - Chain Rule]:
   `\displaystyle du = ((3x)')/(3x) \ dx`
3. [du] Basic Power Rule [Derivative Rule - Multiplied Constant]:
   `\displaystyle du = (3)/(3x) \ dx`
4. [du] Simplify:
   `\displaystyle du = (1)/(x) \ dx`
5. Set dv:
   `\displaystyle dv = (2x - 1) \ dx`
6. [dv] Integration Property [Addition/Subtraction]:
   `\displaystyle v = \int {2x} \, dx - \int {} \, dx`
7. [dv] Integration Property [Multiplied Constant]:
   `\displaystyle v = 2 \int {x} \, dx - \int {} \, dx`
8. [dv] Integration Rule [Reverse Power Rule]:
   `\displaystyle v = x^2 - x`

Step 3: Integrate Pt. 2

1. [Integral] Integration by Parts:
   `\displaystyle \int {(2x - 1) \ln(3x)} \, dx = (x^2 - x) \ln(3x) - \int {(x^2 - x)/(x)} \, dx`
2. [Integrand] Simplify:
   `\displaystyle \int {(2x - 1) \ln(3x)} \, dx = (x^2 - x) \ln(3x) - \int {x - 1} \, dx`
3. [Integral] Rewrite [Integration Property - Addition/Subtraction]:
   `\displaystyle \int {(2x - 1) \ln(3x)} \, dx = (x^2 - x) \ln(3x) - \bigg( \int {x} \, dx - \int {} \, dx \bigg)`
4. [Integral] Integration Rule [Reverse Power Rule]:
   `\displaystyle \int {(2x - 1) \ln(3x)} \, dx = (x^2 - x) \ln(3x) - \bigg( (x^2)/(2) - x \bigg) + C`

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

Unit: Integration