Question

Use integration by parts to find the integrals in Exercise.
∫^0_1 ln 3x dx.

Answer 1

`\displaystyle \int\limits^1_0 {\ln(3x)} \, dx = \ln(3) - 1`

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

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`

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 {\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 Property - Multiplied Constant]:
   `\displaystyle du = (3)/(3x) \ dx`
4. [du] Simplify:
   `\displaystyle du = (1)/(x) \ dx`
5. Set dv:
   `\displaystyle dv = 1 \ dx`
6. [dv] Integration Rule [Reverse Power Rule]:
   `\displaystyle v = x`

Step 3: Integrate Pt. 2

1. [Integral] Integration by Parts:
   `\displaystyle \int\limits^1_0 {\ln(3x)} \, dx = x \ln(3x) \bigg| \limits^1_0 - \int\limits^1_0 {1} \, dx`
2. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int\limits^1_0 {\ln(3x)} \, dx = x \ln(3x) \bigg| \limits^1_0 - \int\limits^1_0 {} \, dx`
3. [Integral] Integration Rule [Reverse Power Rule]:
   `\displaystyle \int\limits^1_0 {\ln(3x)} \, dx = x \ln(3x) \bigg| \limits^1_0 - x \bigg| \limits^1_0`
4. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
   `\displaystyle \int\limits^1_0 {\ln(3x)} \, dx = \ln(3) - 1`

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

Unit: Integration