Evaluate the integrals. ∫1^2 (8 d x)/(x^2-2 x+2)

Name: Problem 73, Chapter 7: Thomas Calculus in SI Units
Uploaded: 2023-03-25T06:17:13-08:00
Duration: 1 min 48 s
Channel: Khushbu Rani
Description: Evaluate the integrals. ∫1^2 (8 d x)/(x^2-2 x+2)

Key Concepts

Factorization of Polynomials

Factorization is the process of expressing a polynomial as a product of its factors. In the context of integrating rational functions, factoring the denominator is crucial because it facilitates the partial fraction decomposition by identifying the simpler polynomial components that comprise the original denominator.

Integration of Rational Functions

This concept involves integrating functions that are ratios of polynomials. The overall strategy typically includes simplifying the integrand through techniques like polynomial division and partial fractions, followed by term-by-term integration using standard antiderivatives for polynomial and logarithmic functions.

Polynomial Long Division

When integrating rational functions where the degree of the numerator is greater than or equal to the degree of the denominator, polynomial long division is used to rewrite the expression as a polynomial plus a proper rational function. This simplifies the integration process by reducing the integrand into a sum of simpler components that can be integrated individually.

Partial Fraction Decomposition

After applying polynomial long division, the remaining proper rational function can often be expressed as a sum of simpler fractions using partial fraction decomposition. This method involves expressing the rational function as a sum of terms with denominators that are either linear or irreducible quadratics, making it easier to integrate each term separately.

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