Final answer:
To integrate ∫(2x-5)/(x²+4x+5) dx, complete the square in the denominator and make the appropriate substitution. The integral simplifies to (1/2)(2 ln|u²+1|-9 arctan(u²+1)) + C.
Step-by-step explanation:
To integrate the expression ∫(2x-5)/(x²+4x+5) dx, we can complete the square in the denominator. The quadratic expression x²+4x+5 can be rewritten as (x+2)²+1. We make the substitution u = x+2, so du = dx.
Now, our integral becomes ∫(2(u-2)-5)/(u²+1) du. Expanding and simplifying the numerator, we have ∫(2u-9)/(u²+1) du.
To integrate this expression, we use the u-substitution method. Let v = u²+1, so dv = 2u du. We can rewrite the integral as (1/2)∫(2u-9)/(u²+1) du = (1/2)∫(2u-9)/v dv. This integral can be simplified further to (1/2)∫(2/v)-9/v dv = (1/2)(2 ln|v|-9 arctan(v)) + C, where C is the constant of integration.