Find solutions to your problems with the help of IDNLearn.com's knowledgeable users. Our platform offers detailed and accurate responses from experts, helping you navigate any topic with confidence.

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]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! IDNLearn.com is your go-to source for accurate answers. Thanks for stopping by, and come back for more helpful information.