Solving the Integration Problem of Rational Functions

The given problem involves the integration of a complex rational function. As a SEO expert, I aim to break down this problem step by step, making it accessible for learners and search engines alike.

Background and Context

The original problem presented is an integral in Calculus, specifically dealing with the integration of a rational function. The rational function to be integrated has a degree in the numerator that is one less than the degree in the denominator. This problem requires the method of solving rational functions and substitution.

Step-by-step Solution

To solve this problem, we will follow a systematic approach. Let's break down our steps in detail:

Step 1: Simplify the Integral

Let's first simplify our integral:

[ I displaystyle int frac{x^3x^2x1}{x^22x^2-3} cdot dx ]

After simplifying the numerator, we get:

[ I displaystyle int frac{x^3x^2x1}{x^22x^2-3} cdot dx displaystyle int frac{1x1x^2}{x^22x^2-3} cdot dx ]

Step 2: Further Simplification

We now apply further algebraic manipulation to decompose the integrand into simpler fractions:

[ I frac{1}{5} cdot displaystyle int frac{(4x x^2)}{x^22x^2-3} cdot dx ]

Further simplifying the expression inside the integral, we can distribute and split the integral as follows:

[ I frac{1}{5} cdot displaystyle int left( frac{4}{x^2-3} frac{x}{x^22} right) dx ]

Step 3: Integration of Each Term

The integral now consists of two simpler integrals, each of which can be solved using standard techniques:

[ I frac{1}{5} cdot left( displaystyle int frac{4}{x^2-3} dx displaystyle int frac{x}{x^22} dx right) ]

Let's integrate each term separately.

[ displaystyle int frac{4}{x^2-3} dx ]

This involves the formula for integrating a form involving ( ln(x^2 - a) ):

[ frac{4}{5} cdot frac{2}{sqrt{3}} ln left| frac{x - sqrt{3}}{x sqrt{3}} right| C_1 ]

And for the second integral:

[ displaystyle int frac{x}{x^22} dx ]

This is a substitution of ( u x^2 2 ), leading to:

[ frac{1}{10} tan^{-1} left( frac{x}{sqrt{2}} right) C_2 ]

Final Solution

Combining the results from the two integrals, we get:

[ I frac{1}{5} left( frac{4}{sqrt{3}} ln left| frac{x - sqrt{3}}{x sqrt{3}} right| 2 ln |x^2 - 3| frac{1}{sqrt{2}} tan^{-1} left( frac{x}{sqrt{2}} right) D right) ]

where ( D ) is the constant of integration.

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