Math Problem Statement

Evaluate the integral \( \int_{e^5}^e \frac{1}{x \sqrt{\ln x}} \, dx \).

Solution

The problem involves evaluating the definite integral:

e5e1xlnxdx.\int_{e^5}^e \frac{1}{x \sqrt{\ln x}} \, dx.

Let us solve this step by step.

Step 1: Substitution

We notice the term lnx\ln x in the denominator. A substitution can simplify this integral: u=lnx    du=1xdx.u = \ln x \implies du = \frac{1}{x} dx. The integral becomes: e5e1xlnxdx=511udu,\int_{e^5}^e \frac{1}{x \sqrt{\ln x}} \, dx = \int_{5}^1 \frac{1}{\sqrt{u}} \, du, where the limits of integration change because u=lnxu = \ln x:

  • When x=e5x = e^5, u=ln(e5)=5u = \ln(e^5) = 5,
  • When x=ex = e, u=ln(e)=1u = \ln(e) = 1.

Thus, the integral simplifies to: 51u1/2du.\int_{5}^1 u^{-1/2} \, du.

Step 2: Evaluate the Integral

The antiderivative of u1/2u^{-1/2} (or 1u\frac{1}{\sqrt{u}}) is: u1/2du=2u1/2.\int u^{-1/2} \, du = 2u^{1/2}. Applying this to the integral: 51u1/2du=2u1/251.\int_{5}^1 u^{-1/2} \, du = 2u^{1/2} \Big|_{5}^1.

Step 3: Substitute the Limits

Substitute u=1u = 1 and u=5u = 5: 2u1/251=2125=225.2u^{1/2} \Big|_{5}^1 = 2\sqrt{1} - 2\sqrt{5} = 2 - 2\sqrt{5}.

Final Answer:

225\boxed{2 - 2\sqrt{5}}

Would you like me to provide additional details or clarify any step?


Here are 5 related questions to deepen your understanding:

  1. Why is substitution a good method for this type of integral?
  2. How do the limits of integration change during substitution?
  3. What is the geometric interpretation of this definite integral?
  4. Can you generalize this integral for different powers of lnx\ln x?
  5. What happens if the integral's bounds are flipped?

Tip: Always double-check the new integration limits when performing substitution in definite integrals!

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Math Problem Analysis

Mathematical Concepts

Integration
Substitution Method
Definite Integrals

Formulas

\( \int u^{-1/2} \, du = 2u^{1/2} + C \)

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

College Calculus or Advanced High School Calculus