Answer:
[tex]\displaystyle \int {\log x} \, dx = x \bigg( \log x - \frac{1}{\ln(10)} \bigg) + C[/tex]
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
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]
Integration by Parts: [tex]\displaystyle \int {u} \, dv = uv - \int {v} \, du[/tex]
- [IBP] LIPET: Logs, inverses, Polynomials, Exponentials, Trig
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle \int {\log x} \, dx[/tex]
Step 2: Integrate Pt. 1
Identify variables for integration by parts using LIPET.
- Set u: [tex]\displaystyle u = \log x[/tex]
- [u] Logarithmic Differentiation: [tex]\displaystyle du = \frac{1}{x \ln(10)} \ dx[/tex]
- Set dv: [tex]\displaystyle dv = dx[/tex]
- [dv] Integration Rule [Reverse Power Rule]: [tex]\displaystyle v = x[/tex]
Step 3: integrate Pt. 2
- [Integral] Integration by Parts: [tex]\displaystyle \int {\log x} \, dx = x \log x - \int {\frac{1}{\ln(10)}} \, dx[/tex]
- [Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle \int {\log x} \, dx = x \log x - \frac{1}{\ln(10)} \int {} \, dx[/tex]
- [Integral] Integration Rule [Reverse Power Rule]: [tex]\displaystyle \int {\log x} \, dx = x \log x - \frac{1}{\ln(10)}x + C[/tex]
- Factor: [tex]\displaystyle \int {\log x} \, dx = x \bigg( \log x - \frac{1}{\ln(10)} \bigg) + C[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
