Sure, let’s integrate the function
[tex]\[ \left(x^2 + x - 2\right)^{-\frac{1}{2}} \][/tex]
To integrate the function [tex]\( \left(x^2 + x - 2\right)^{-\frac{1}{2}} \)[/tex] with respect to [tex]\( x \)[/tex], we can follow these steps:
1. Identify the integrand:
[tex]\[ \left(x^2 + x - 2\right)^{-\frac{1}{2}} \][/tex]
2. Factor the quadratic expression inside the integrand:
[tex]\[ x^2 + x - 2 = (x + 2)(x - 1) \][/tex]
So,
[tex]\[ \left(x^2 + x - 2\right)^{-\frac{1}{2}} = \left[(x + 2)(x - 1)\right]^{-\frac{1}{2}} \][/tex]
3. Simplify the integrand:
[tex]\[ \left[(x + 2)(x - 1)\right]^{-\frac{1}{2}} = (x + 2)^{-\frac{1}{2}} (x - 1)^{-\frac{1}{2}} \][/tex]
4. Use substitution:
Let's use the substitution method to make integration easier. Set
[tex]\[ u = x^2 + x - 2 \][/tex]
Then, the derivative is:
[tex]\[ \frac{du}{dx} = 2x + 1 \][/tex]
Hence,
[tex]\[ du = (2x + 1)dx \][/tex]
5. Adjust the integrand to match the form involving [tex]\(du\)[/tex]:
Notice that we can manipulate the integrand to effectively utilize [tex]\(du\)[/tex].
[tex]\[
\int (x^2 + x - 2)^{-\frac{1}{2}} dx
\][/tex]
6. Identify the antiderivative:
The antiderivative of [tex]\(\left(x^2 + x - 2\right)^{-\frac{1}{2}}\)[/tex] ends up being:
[tex]\[
\ln(2x + 2\sqrt{x^2 + x - 2} + 1)
\][/tex]
Therefore, the integral of the function [tex]\( \left(x^2 + x - 2\right)^{-\frac{1}{2}} \)[/tex] with respect to [tex]\( x \)[/tex] can be expressed as:
[tex]\[
\int \left(x^2 + x - 2\right)^{-\frac{1}{2}} \, dx = \ln\left(2x + 2\sqrt{x^2 + x - 2} + 1\right) + C
\][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
