Math Problem Statement

সমাকলন কর: ∫(e^x/x)(1 + x ln x) dx.

Solution

To solve the given integral:

exx(1+xlnx)dx\int \frac{e^x}{x} \left( 1 + x \ln x \right) dx

We will approach this step by step:

Step 1: Break Down the Expression

The integral can be separated into two parts: exx(1+xlnx)dx=exxdx+exlnxdx.\int \frac{e^x}{x} \left( 1 + x \ln x \right) dx = \int \frac{e^x}{x} dx + \int e^x \ln x \, dx.

Step 2: Solve Each Term

Term 1: exxdx\int \frac{e^x}{x} dx

The integral exxdx\int \frac{e^x}{x} dx does not have a solution in terms of elementary functions. This is known as the exponential integral and is typically expressed using the special function Ei(x)\text{Ei}(x).

Term 2: exlnxdx\int e^x \ln x \, dx

This requires integration by parts:

  • Let u=lnxu = \ln x, so that du=1xdxdu = \frac{1}{x} dx.
  • Let dv=exdxdv = e^x dx, so that v=exv = e^x.

Using the integration by parts formula: udv=uvvdu,\int u \, dv = uv - \int v \, du, we get: exlnxdx=exlnxex1xdx.\int e^x \ln x \, dx = e^x \ln x - \int e^x \cdot \frac{1}{x} dx.

Simplified Result

Thus, the integral becomes: exx(1+xlnx)dx=Ei(x)+(exlnxEi(x)).\int \frac{e^x}{x} \left( 1 + x \ln x \right) dx = \text{Ei}(x) + \left( e^x \ln x - \text{Ei}(x) \right).

Combining terms, the final solution is: exx(1+xlnx)dx=exlnx.\int \frac{e^x}{x} \left( 1 + x \ln x \right) dx = e^x \ln x.

Let me know if you'd like further clarifications or step-by-step details!


Related Questions:

  1. What is the definition and significance of the exponential integral function, Ei(x)\text{Ei}(x)?
  2. How do we apply integration by parts for functions involving logarithms and exponentials?
  3. Can exxdx\int \frac{e^x}{x} dx be solved numerically? If so, how?
  4. What are the limitations of expressing certain integrals in terms of elementary functions?
  5. How does the substitution method compare to integration by parts in solving complex integrals?

Tip:

Always check if a special function or numerical approach is required for integrals involving expressions like exx\frac{e^x}{x}. This avoids unnecessary complications in manual computation.

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Math Problem Analysis

Mathematical Concepts

Calculus
Integration
Exponential and Logarithmic Functions

Formulas

Integration by parts: ∫u dv = uv - ∫v du
Exponential integral function: Ei(x)

Theorems

Integration by parts theorem

Suitable Grade Level

Undergraduate Mathematics