Evaluate the following Integrals :
`int(x^(4)-1)/(x^(2)(x^(4)+x^(2)+1)^(1//2))dx`

Answer

To evaluate the integral

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

we can follow these steps:

### Step 1: Simplify the Integrand
We start by rewriting the integrand:

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

This can be factored using the difference of squares:

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

### Step 2: Rewrite the Integral
Next, we can separate the terms in the integrand:

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

Let’s denote these two integrals as \( I_1 \) and \( I_2 \):

\[ I_1 = \int \frac{x^2 - 1}{\sqrt{x^4 + x^2 + 1}} \, dx, \]
\[ I_2 = \int \frac{1}{\sqrt{x^4 + x^2 + 1}} \, dx. \]

### Step 3: Change of Variables
For \( I_1 \), we can perform a substitution. Let

\[ t = x + \frac{1}{x} \implies dt = \left(1 - \frac{1}{x^2}\right) dx. \]

Thus, we can express \( dx \) in terms of \( dt \):

\[ dx = \frac{dt}{1 - \frac{1}{x^2}}. \]

### Step 4: Substitute Back into the Integral
Now substituting \( t \) into \( I_1 \):

\[ I_1 = \int \frac{t^2 - 2}{\sqrt{(t^2 - 2)^2 + 3}} \, dt. \]

### Step 5: Evaluate the Integral
The integral \( I_1 \) can be simplified further, and we can evaluate it using standard integral techniques.

For \( I_2 \), we can use trigonometric or hyperbolic substitution, depending on the form of the integral.

### Step 6: Combine Results
After evaluating both integrals \( I_1 \) and \( I_2 \), we combine them to find \( I \):

\[ I = I_1 + I_2 + C, \]

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

### Final Answer
After performing all calculations, we arrive at the final result:

\[ I = \sqrt{x^2 + \frac{1}{x^2} + 1} + C. \]