Math Problem Statement

1/x(lnx)^2

Solution

To solve the integral 1x(lnx)2dx\int \frac{1}{x (\ln x)^2} \, dx, we can use substitution.

Step-by-Step Solution:

  1. Substitute u=lnxu = \ln x.
    Then, du=1xdxdu = \frac{1}{x} \, dx, so dx=xdudx = x \, du.

  2. Rewrite the integral: 1x(lnx)2dx=1u2du\int \frac{1}{x (\ln x)^2} \, dx = \int \frac{1}{u^2} \cdot du

  3. Integrate 1u2du\int \frac{1}{u^2} \, du: The integral of 1u2\frac{1}{u^2} is 1u-\frac{1}{u}.

    Therefore: 1u+C-\frac{1}{u} + C

  4. Substitute back u=lnxu = \ln x: 1lnx+C-\frac{1}{\ln x} + C

Final Answer:

1x(lnx)2dx=1lnx+C\int \frac{1}{x (\ln x)^2} \, dx = -\frac{1}{\ln x} + C

Would you like any more details, or do you have any questions?

Here are 5 related questions to further your understanding:

  1. How would you solve 1xlnxdx\int \frac{1}{x \ln x} \, dx?
  2. What is the integral 1(lnx)2dx\int \frac{1}{(\ln x)^2} \, dx?
  3. How would the solution change if the integrand were 1x(lnx)3\frac{1}{x (\ln x)^3}?
  4. Can you evaluate x2lnxdx\int x^2 \ln x \, dx using integration by parts?
  5. What happens if you try to integrate 1x2(lnx)2dx\int \frac{1}{x^2 (\ln x)^2} \, dx?

Tip: Substitution is a powerful technique when you see a function and its derivative present in the integrand.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Calculus
Integration
Substitution

Formulas

Integral substitution formula

Theorems

-

Suitable Grade Level

College