Evaluate the given definite integrals. ∫-1^2 (8 x-2)/((2 x^2-x+1)^3) d x

Name: Problem 29, Chapter 25: Basic Technical Mathematics with Calculus
Uploaded: 2021-06-06T07:31:32-08:00
Duration: 3 min 22 s
Channel: Dwijendra Rao
Description: Evaluate the given definite integrals. ∫-1^2 (8 x-2)/((2 x^2-x+1)^3) d x

step 1 Hello, we have to evaluate the given definite integral that is given minus 1 to 2, 8 x minus 2 u

Evaluate the given definite integrals. ∫-1^2 (8 x-2)/((2...

Key Concepts

Definite Integrals

Definite integrals represent the net accumulation of a quantity, where the limits of integration specify the starting and ending points. In calculus, evaluating a definite integral involves finding an antiderivative of the integrand and then applying the Fundamental Theorem of Calculus by computing the difference of the antiderivative at the upper and lower bounds.

U-Substitution

U-substitution is a technique used to simplify integration by changing variables. It is especially useful when the integrand contains a composite function whose inner function’s derivative is present in the integrand. This method transforms the original integral into an easier form by substituting part of the expression with a new variable, and it requires adjusting the limits of integration when dealing with definite integrals.

Chain Rule

The chain rule is a fundamental concept in differentiation that is closely related to u-substitution in integration. It explains how to differentiate composite functions, and conversely, it informs the process of integrating functions where the derivative of an inner function is present in the integrand. In the context of integration, recognizing a pattern that reflects the chain rule can simplify the problem significantly.

Integration of Rational Functions

Integration of rational functions involves finding antiderivatives for functions that are expressed as ratios of polynomials. Techniques such as u-substitution, partial fractions, or algebraic manipulation are applied depending on the structure of the numerator and denominator. Recognizing when a substitution can simplify the integral is key in handling these types of problems.

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