Math Problem Statement

how to integrate (1 + lnx) * x^x

Solution

To integrate the function (1+lnx)xx(1 + \ln x) \cdot x^x, we use substitution and advanced techniques. Here’s the step-by-step approach:

Step 1: Rewrite the function

The integral is: I=(1+lnx)xxdxI = \int (1 + \ln x) \cdot x^x \, dx Let’s recognize that xx=exlnxx^x = e^{x \ln x}, and rewrite the function: I=(1+lnx)exlnxdxI = \int (1 + \ln x) \cdot e^{x \ln x} \, dx

Step 2: Substitution

Set u=xx=exlnxu = x^x = e^{x \ln x}. Then: lnu=xlnx\ln u = x \ln x

Differentiating u=xxu = x^x, we get: dudx=xx(1+lnx)\frac{du}{dx} = x^x (1 + \ln x) Thus: du=xx(1+lnx)dxdu = x^x (1 + \ln x) \, dx

This substitution simplifies the integral: I=duI = \int du

Step 3: Solve the simplified integral

The integral of dudu is: I=u+CI = u + C

Step 4: Substitute back

Since u=xxu = x^x, the final solution is: I=xx+CI = x^x + C

Final Answer:

(1+lnx)xxdx=xx+C\int (1 + \ln x) \cdot x^x \, dx = x^x + C


Do you want a deeper explanation or have additional questions?

Here are five related questions:

  1. How does the derivative of xxx^x work?
  2. Can this integral be solved without substitution?
  3. What happens if we integrate just xxx^x without the 1+lnx1 + \ln x?
  4. Can you evaluate this integral for specific limits (e.g., from 1 to 2)?
  5. How can we verify the solution by differentiation?

Tip: When working with exponential functions like xxx^x, rewriting them in terms of ef(x)e^{f(x)} 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)