Math Problem Statement
Solve the integral \( \int \frac{\ln x}{x^2} \, dx \)
Solution
The integral given in the image is:
To solve this integral, we can use the method of integration by parts. Recall the formula for integration by parts:
Let , so that .
Let , so that .
Now apply the formula:
Substitute , , and :
Simplify:
The second integral is straightforward:
Now substitute back:
Final answer:
Let me know if you'd like further clarification or details on this solution.
Here are 5 related questions for further exploration:
- How does integration by parts work for different types of functions?
- What are some practical applications of logarithmic integrals in physics or engineering?
- How can substitution simplify other integrals with logarithmic terms?
- How does the constant of integration influence the solution?
- What happens if we change the power of in the denominator?
Tip: When choosing in integration by parts, pick the function that simplifies upon differentiation.
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Math Problem Analysis
Mathematical Concepts
Calculus
Integration by Parts
Logarithmic Functions
Formulas
Integration by parts formula: \( \int u \, dv = uv - \int v \, du \)
Theorems
Integration by parts
Suitable Grade Level
College Level