integration of (1+x-1/x)e^(1+1/x)

How can we solve this sum. Could someone please help me?????

To tackle the integration of the expression \( (1 + x - \frac{1}{x}) e^{(1 + \frac{1}{x})} \), we can break it down into manageable parts. The key here is to recognize that the integrand can be simplified using substitution techniques. Let's go through the steps together.

Step 1: Identify a Suitable Substitution

Notice that the term \( e^{(1 + \frac{1}{x})} \) suggests that we might benefit from substituting \( u = 1 + \frac{1}{x} \). This substitution will help simplify the exponent and the overall expression.

Finding the Derivative

To use this substitution effectively, we need to express \( dx \) in terms of \( du \). First, we differentiate \( u \):

• Since \( u = 1 + \frac{1}{x} \), we can rewrite it as \( u - 1 = \frac{1}{x} \).
• Taking the derivative gives us \( du = -\frac{1}{x^2} dx \), or rearranging, \( dx = -x^2 du \).

Step 2: Express \( x \) in Terms of \( u \)

From our substitution, we have \( x = \frac{1}{u - 1} \). This will allow us to express all parts of the integrand in terms of \( u \).

Rewriting the Integrand

Now, let's rewrite \( (1 + x - \frac{1}{x}) e^{(1 + \frac{1}{x})} \):

• Substituting \( x \) gives us \( 1 + \frac{1}{u - 1} - (u - 1) \).
• Combining these terms leads to \( 1 + \frac{1}{u - 1} - u + 1 = 2 - u + \frac{1}{u - 1} \).

Step 3: Integrate the New Expression

Now we can express the integral in terms of \( u \):

We have:

\( \int (2 - u + \frac{1}{u - 1}) e^u (-\frac{1}{(u - 1)^2}) du \).

This integral can be split into simpler parts, allowing us to integrate each term separately.

Breaking Down the Integral

We can separate the integral into three parts:

• \( -2 \int \frac{e^u}{(u - 1)^2} du \)
• \( + \int \frac{u e^u}{(u - 1)^2} du \)
• \( -\int \frac{e^u}{(u - 1)^2} du \)

Step 4: Solve Each Integral

Each of these integrals can be approached using integration by parts or other techniques, depending on their complexity. For example, the integral of \( e^u \) is straightforward, while the others may require more advanced methods.

Final Steps

After integrating each part, remember to substitute back \( u = 1 + \frac{1}{x} \) to express the final answer in terms of \( x \). This will give you the complete solution to the original integral.

In summary, the integration process involves recognizing suitable substitutions, rewriting the integrand, and breaking down the integral into manageable parts. With practice, these techniques will become more intuitive, allowing you to tackle even more complex integrals with confidence.