Math Problem Statement
how to integrate (1 + lnx) * x^x
Solution
To integrate the function , we use substitution and advanced techniques. Here’s the step-by-step approach:
Step 1: Rewrite the function
The integral is: Let’s recognize that , and rewrite the function:
Step 2: Substitution
Set . Then:
Differentiating , we get: Thus:
This substitution simplifies the integral:
Step 3: Solve the simplified integral
The integral of is:
Step 4: Substitute back
Since , the final solution is:
Final Answer:
Do you want a deeper explanation or have additional questions?
Here are five related questions:
- How does the derivative of work?
- Can this integral be solved without substitution?
- What happens if we integrate just without the ?
- Can you evaluate this integral for specific limits (e.g., from 1 to 2)?
- How can we verify the solution by differentiation?
Tip: When working with exponential functions like , rewriting them in terms of often simplifies differentiation and integration.
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Math Problem Analysis
Mathematical Concepts
Integration
Exponential functions
Substitution in integration
Formulas
x^x = e^{x ln x}
du = x^x (1 + ln x) dx
∫ du = u + C
Theorems
Fundamental Theorem of Calculus
Logarithmic and Exponential Properties
Suitable Grade Level
College Level (Calculus II)