The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫₁² (8 dx / (x² - 2x + 2))

Verified step by step guidance

1. Start by examining the integral \( \int_1^2 \frac{8}{x^2 - 2x + 2} \, dx \). Notice that the denominator is a quadratic expression that can be rewritten to complete the square.
2. Rewrite the quadratic in the denominator by completing the square: \( x^2 - 2x + 2 = (x - 1)^2 + 1 \). This form is easier to work with for integration.
3. Make the substitution \( u = x - 1 \), which implies \( du = dx \). Change the limits of integration accordingly: when \( x = 1 \), \( u = 0 \); when \( x = 2 \), \( u = 1 \). The integral becomes \( \int_0^1 \frac{8}{u^2 + 1} \, du \).
4. Recognize that the integral \( \int \frac{1}{u^2 + 1} \, du \) corresponds to the inverse tangent function \( \arctan(u) \). Use this fact to express the integral in terms of \( \arctan(u) \).
5. Integrate using the formula \( \int \frac{1}{u^2 + a^2} \, du = \frac{1}{a} \arctan\left( \frac{u}{a} \right) + C \). Since \( a = 1 \), the integral simplifies to \( 8 \arctan(u) \) evaluated from \( u = 0 \) to \( u = 1 \).

Key Concepts

Integration of Rational Functions

This involves integrating functions expressed as ratios of polynomials. Techniques often include algebraic manipulation such as completing the square or partial fraction decomposition to simplify the integrand before integration.

Completing the Square

Completing the square rewrites a quadratic expression into the form (x - h)² + k, which can simplify the integral, especially when the denominator is a quadratic. This form often allows the use of standard integral formulas involving inverse trigonometric functions.

Integration Using Trigonometric Substitution

When the integrand contains expressions like a² + x² or (x - h)² + k, trigonometric substitution can transform the integral into a simpler trigonometric integral. This method leverages identities such as 1 + tan²θ = sec²θ to facilitate integration.