Math Problem Statement
Solution
The integral presented is:
This type of integral is related to the differentiation of logarithmic functions. Let's solve it step by step.
Solution:
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Substitution: Let . Then: Also, when , , and when , .
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Rewrite the integral: Substituting and :
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Evaluate the integral: The integral of is: Apply the limits and :
Final Answer:
Would you like a detailed explanation of any specific step or more examples related to this problem?
Here are 5 related questions for deeper understanding:
- How does substitution simplify solving integrals involving logarithms?
- Can you solve using substitution?
- How would the answer change if the upper limit was instead of ?
- What other functions are commonly integrated using substitution?
- Can this integral be solved using integration by parts? Why or why not?
Tip: When encountering integrals involving , look for patterns involving its derivative , as substitution often simplifies these problems significantly.
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Math Problem Analysis
Mathematical Concepts
Integration
Logarithmic Functions
Substitution Method
Formulas
\( \int \frac{\ln x}{x} \, dx = \frac{(\ln x)^2}{2} + C \)
Substitution: \( u = \ln x \), \( du = \frac{1}{x} dx \)
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12 or early college
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