Evaluate the following integration
`int((1+x)^(2))/(x(1+x^(2)))dx`

Answer

To evaluate the integral

\[ I = \int \frac{(1+x)^2}{x(1+x^2)} \, dx, \]

we will follow these steps:

### Step 1: Expand the numerator
Using the formula \((A + B)^2 = A^2 + B^2 + 2AB\), we can expand \((1 + x)^2\):

\[ (1 + x)^2 = 1^2 + x^2 + 2 \cdot 1 \cdot x = 1 + x^2 + 2x. \]

So we can rewrite the integral as:

\[ I = \int \frac{1 + x^2 + 2x}{x(1 + x^2)} \, dx. \]

### Step 2: Split the integral
Now, we can split the integral into three separate integrals:

\[ I = \int \frac{1}{x(1 + x^2)} \, dx + \int \frac{x^2}{x(1 + x^2)} \, dx + 2 \int \frac{x}{x(1 + x^2)} \, dx. \]

This simplifies to:

\[ I = \int \frac{1}{x(1 + x^2)} \, dx + \int \frac{x}{1 + x^2} \, dx + 2 \int \frac{1}{1 + x^2} \, dx. \]

### Step 3: Simplify the integrals
The second integral simplifies as follows:

\[ \int \frac{x}{1 + x^2} \, dx = \frac{1}{2} \ln(1 + x^2) + C_1, \]

and the third integral is a standard integral:

\[ \int \frac{1}{1 + x^2} \, dx = \tan^{-1}(x) + C_2. \]

### Step 4: Solve the first integral
For the first integral, we can use partial fraction decomposition:

\[ \frac{1}{x(1 + x^2)} = \frac{A}{x} + \frac{Bx + C}{1 + x^2}. \]

Multiplying through by the denominator \(x(1 + x^2)\) and equating coefficients, we can solve for \(A\), \(B\), and \(C\). After finding these constants, we can integrate each term.

### Step 5: Combine the results
After computing each integral, we combine the results:

\[ I = \ln|x| + \frac{1}{2} \ln(1 + x^2) + 2 \tan^{-1}(x) + C, \]

where \(C\) is the constant of integration.

### Final Result
Thus, the final result of the integral is:

\[ I = \ln|x| + \frac{1}{2} \ln(1 + x^2) + 2 \tan^{-1}(x) + C. \]

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