Evaluate the following integration
`int(x^(4)+x^(2)+1)/(2(1+x^(2)))dx`

Answer

To evaluate the integral

\[ I = \int \frac{x^4 + x^2 + 1}{2(1 + x^2)} \, dx, \]

we will follow these steps:

### Step 1: Rewrite the Integral
We can rewrite the integral as:

\[ I = \frac{1}{2} \int \frac{x^4 + x^2 + 1}{1 + x^2} \, dx. \]

### Step 2: Simplify the Integrand
Next, we can simplify the integrand by dividing \(x^4 + x^2 + 1\) by \(1 + x^2\). We can do polynomial long division or notice that:

\[ x^4 + x^2 + 1 = (x^2)(1 + x^2) + 1. \]

Thus, we can rewrite the integrand as:

\[ \frac{x^4 + x^2 + 1}{1 + x^2} = x^2 + \frac{1}{1 + x^2}. \]

### Step 3: Substitute Back into the Integral
Now, substituting this back into the integral, we have:

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

### Step 4: Separate the Integral
We can separate the integral into two parts:

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

### Step 5: Evaluate Each Integral
Now we evaluate each integral separately.

1. The integral of \(x^2\) is:

\[ \int x^2 \, dx = \frac{x^3}{3}. \]

2. The integral of \(\frac{1}{1 + x^2}\) is a standard integral:

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

### Step 6: Combine the Results
Now substituting back into our expression for \(I\):

\[ I = \frac{1}{2} \left( \frac{x^3}{3} + \tan^{-1}(x) \right) + C, \]

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

### Final Answer
Thus, the final result is:

\[ I = \frac{x^3}{6} + \frac{1}{2} \tan^{-1}(x) + C. \]

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