Evaluate the integral. \[ \int_{0}^{1} \frac{x-1}{x^{2}-2 x+5} d x \] $\int_{0}^{1} \frac{x-1}{x^{2}-2 x+5} d x \approx$ (Type an integer or decimal rounded to three decimal places as needed.)
Solution
Step 1 ：The integral is a definite integral from 0 to 1. The integrand is a rational function.
Step 2 ：We can use numerical methods to approximate the value of the integral. One common method is Simpson's rule, which approximates the area under the curve by dividing it into a number of equal-width intervals and approximating the area of each interval with a parabola.
Step 3 ：We set n = 1000 and h = 0.001.
Step 4 ：We calculate the x values from 0 to 1 with a step of 0.001, and the corresponding y values using the function \(\frac{x-1}{x^{2}-2 x+5}\).
Step 5 ：We then calculate the integral using Simpson's rule and get an approximate value of -0.11157177565710732.
Step 6 ：The approximate value of the integral \(\int_{0}^{1} \frac{x-1}{x^{2}-2 x+5} d x\) is \(\boxed{-0.112}\).
