At IDNLearn.com, find answers to your most pressing questions from experts and enthusiasts alike. Our experts are available to provide in-depth and trustworthy answers to any questions you may have.

8. Evaluate the integral:
[tex]\[ \int \frac{x^2 - x - 2}{1 - x^2} \, dx \][/tex]


Sagot :

To solve the integral [tex]\[ \int \frac{x^2 - x - 2}{1 - x^2} \, dx, \][/tex] we can follow these steps:

1. Simplify the integrand:
Notice that the denominator [tex]\( 1 - x^2 \)[/tex] can be factored as [tex]\( (1 - x)(1 + x) \)[/tex].

The integrand becomes:
[tex]\[ \frac{x^2 - x - 2}{1 - x^2} = \frac{x^2 - x - 2}{(1 - x)(1 + x)}. \][/tex]

Simplifying the numerator [tex]\(x^2 - x - 2\)[/tex], we can factor it as [tex]\( (x - 2)(x + 1) \)[/tex].

So, the integrand can be rewritten as:
[tex]\[ \frac{(x - 2)(x + 1)}{(1 - x)(1 + x)}. \][/tex]

2. Simplify further:
Notice that [tex]\( (1 - x) \)[/tex] can be written as [tex]\(-(x - 1)\)[/tex]:
[tex]\[ \frac{(x - 2)(x + 1)}{(1 - x)(1 + x)} = \frac{(x - 2)(x + 1)}{-[-(x - 1)](x + 1)} = -\frac{(x - 2)(x + 1)}{(x - 1)(x + 1)}. \][/tex]

The [tex]\((x + 1)\)[/tex] terms cancel out:
[tex]\[ -\frac{(x - 2)}{(x - 1)}. \][/tex]

So the integrand simplifies to:
[tex]\[ \int -\frac{x - 2}{x - 1} \, dx \][/tex]

3. Split the fraction:
We can split the fraction in the integrand:
[tex]\[ -\frac{x - 2}{x - 1} = -\left( \frac{x - 1}{x - 1} + \frac{-1}{x - 1} \right) = -\left( 1 + \frac{-1}{x - 1} \right). \][/tex]

Simplifying further:
[tex]\[ -\left( 1 + \frac{-1}{x - 1} \right) = -1 - \frac{-1}{x - 1} = -1 + \frac{1}{x - 1}. \][/tex]

So the integrand now is:
[tex]\[ \int \left( -1 + \frac{1}{x - 1} \right) \, dx. \][/tex]

4. Integrate term-by-term:
We can integrate each term individually:
[tex]\[ \int \left( -1 + \frac{1}{x - 1} \right) \, dx = \int -1 \, dx + \int \frac{1}{x - 1} \, dx. \][/tex]

The integral of [tex]\(-1\)[/tex] is:
[tex]\[ \int -1 \, dx = -x. \][/tex]

The integral of [tex]\(\frac{1}{x - 1}\)[/tex] is:
[tex]\[ \int \frac{1}{x - 1} \, dx = \log |x - 1|. \][/tex]

Putting everything together:
[tex]\[ -x + \log |x - 1| + C, \][/tex]

where [tex]\(C\)[/tex] is the constant of integration.

Hence, the integral [tex]\[ \int \frac{x^2 - x - 2}{1 - x^2} \, dx \][/tex] evaluates to:
[tex]\[ -x + \log |x - 1| + C. \][/tex]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. For dependable answers, trust IDNLearn.com. Thank you for visiting, and we look forward to assisting you again.