The image shows a transformation of a given integral into a more simplified or decomposed form.

First Integral:

The original integral is:

$\int \frac{1 - \frac{3}{x} + \frac{1}{x^2}}{x^2 + 1 + \frac{1}{x^2}} \, dx$

This is a complex rational expression involving terms with powers of $x$, their reciprocals, and constant terms. The goal is likely to simplify the expression into parts that are easier to integrate individually.

Second Step (Transformed Expression):

The integral has been rewritten in two parts:

$\int \frac{1 + \frac{1}{x^2}}{\left(x - \frac{1}{x}\right)^2 + 3} \, dx - \int \frac{3x}{x^4 + x^2 + 1} \, dx$

Part 1:

The first term:

$\int \frac{1 + \frac{1}{x^2}}{\left(x - \frac{1}{x}\right)^2 + 3} \, dx$

This suggests a substitution was applied to transform the original rational expression into a form resembling a standard trigonometric integral (related to arc functions or trigonometric identities). The expression $\left( x - \frac{1}{x} \right)$ indicates the substitution might involve transforming the variable $x$ to a new function that simplifies the denominator, likely leading to the appearance of a trigonometric identity like $\tan^{-1}$.

Part 2:

The second integral:

$\int \frac{3x}{x^4 + x^2 + 1} \, dx$

This term is isolated and represents a separate integral to evaluate. The structure of the denominator $x^4 + x^2 + 1$ hints that the substitution or method to solve this part might involve partial fractions or other advanced integration techniques.

Summary of Steps:

1. Numerator and Denominator Simplification: The original expression was broken into terms that can be managed using standard substitutions.
2. Substitution and Decomposition: The integral was decomposed into two simpler-looking integrals, with the first one involving the transformation $x - \frac{1}{x}$ and the second one separated as a more standard rational function of $x$.

Would you like a detailed step-by-step explanation for how to solve these two integrals?

Relative Questions:

1. What substitution can be applied to simplify the expression $x - \frac{1}{x}$?
2. How do trigonometric substitutions apply to integrals like the first part?
3. Can the second part of the integral be solved using partial fraction decomposition?
4. What role do inverse trigonometric functions play in integrals of this form?
5. How do you recognize when to split an integral into two parts?

Tip:

When faced with a complex integral, try decomposing the expression into smaller, more manageable parts—look for symmetry or opportunities for substitution to simplify the integration.