Dear sir,

Plz give the solution of this indefinite integral

∫(e^x(1-x))/((1+x^2)^2)dx = ?

To solve the indefinite integral ∫(e^x(1-x))/((1+x^2)^2)dx, we can use integration by parts and substitution. This integral involves a product of functions, which makes it a good candidate for integration by parts. Let's break it down step by step.

Step 1: Identify Parts for Integration by Parts

In integration by parts, we typically use the formula:

∫u dv = uv - ∫v du

For our integral, we can let:

• u = 1 - x (which simplifies when we differentiate it)
• dv = e^x / (1 + x^2)^2 dx

Step 2: Differentiate and Integrate

Now, we need to find du and v:

• du = -dx
• To find v, we need to integrate dv:

Finding v involves integrating e^x / (1 + x^2)^2. This integral can be approached using a substitution method or recognizing it as a standard form. However, it may not yield a simple elementary function. For practical purposes, we can denote this integral as:

v = ∫(e^x / (1 + x^2)^2) dx

Step 3: Apply Integration by Parts

Now we can apply the integration by parts formula:

∫(e^x(1-x))/((1+x^2)^2)dx = (1-x)v - ∫v(-dx)

This simplifies to:

∫(e^x(1-x))/((1+x^2)^2)dx = (1-x)v + ∫v dx

Step 4: Evaluate the Remaining Integral

The remaining integral ∫v dx can be complex, depending on how we express v. If we can find a closed form for v, we can substitute it back into our equation. However, if v does not yield a simple form, we may need to resort to numerical methods or series expansion for practical evaluation.

Step 5: Final Expression

Thus, the final expression for the integral can be summarized as:

∫(e^x(1-x))/((1+x^2)^2)dx = (1-x)∫(e^x / (1 + x^2)^2)dx + C

Where C is the constant of integration. Depending on the context or specific limits, further simplification may be possible, or numerical methods may be employed for evaluation.

In summary, while the integral involves some complex components, breaking it down using integration by parts and recognizing the need for further evaluation of the integral of v allows us to approach the solution systematically. If you have any further questions or need clarification on any step, feel free to ask!