What is the integration of (ln(x))^n?

  • Context: Undergrad 
  • Thread starter Thread starter saeed69
  • Start date Start date
  • Tags Tags
    Integration
Click For Summary
SUMMARY

The integration of (ln(x))^n can be effectively performed using integration by parts. For the first integral, ∫ln(x)dx, the result is xln(x) - x + C. For the second integral, ∫(ln(x))^2dx, the integration by parts yields x(ln(x))^2 - 2x + C. This method can be applied iteratively to higher powers of ln(x) to identify a consistent pattern in the results.

PREREQUISITES
  • Understanding of integration by parts
  • Familiarity with logarithmic functions
  • Basic knowledge of calculus
  • Ability to manipulate algebraic expressions
NEXT STEPS
  • Study advanced integration techniques, including repeated integration by parts
  • Explore the properties of logarithmic functions in calculus
  • Learn about integration of exponential functions
  • Investigate the use of integration by parts in solving differential equations
USEFUL FOR

Students of calculus, mathematics educators, and anyone seeking to master integration techniques involving logarithmic functions.

saeed69
Messages
3
Reaction score
0
what is the integration of (ln(x))^n?
 
Physics news on Phys.org


To integrate ln x by parts, let u= ln x, dv= dx. Then du= (1/x)dx and v= x so
\int ln x= x(ln x)- \int dx= xln x- x+ C

To integrate (ln x)^2 by parts, let u= ln x, dv= ln x dx. Then du= (1/x)dx and v= xln x- x so
x(ln x)^2- x ln x- \int ln x- 1 dx= x(ln x)^2- x ln x- (x ln x - x- x)= x(ln x)^2- 2x+ C

Keep integrating by parts until you see a pattern.
 

Similar threads

Undergrad Integrating 1/x with units (logarithm)
  • · Replies 15 ·
Replies
15
Views
4K
Undergrad Did Math DF's integral calculator make a glaring mistake?
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Why Are There Different Forms of the Integration Formula for Cosecant?
  • · Replies 3 ·
Replies
3
Views
2K
High School Integral of cosecant function: understanding different approaches
  • · Replies 14 ·
Replies
14
Views
4K
Undergrad Verifying Integration of ##\int_0^1 x^m \ln x \, \mathrm{d}x##
  • · Replies 1 ·
Replies
1
Views
2K
High School Why is the definite integral of 1/x from -1 to 1 undefined?
  • · Replies 4 ·
Replies
4
Views
6K
Undergrad Solving the Difficult Integral ##\int_0^{\infty} x^{n+1} e^{-x} \sin(ax) dx##
  • · Replies 5 ·
Replies
5
Views
3K
Graduate How do I apply the method of steepest descents to this integral?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Having trouble understanding a seemingly simple u-substitution
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Integration of a hyperbolic function
  • · Replies 5 ·
Replies
5
Views
2K