Question

$19-38$ Evaluate the integral. $$\int_{-1}^{2}(3 u-2)(u+1) d u$$

   $19-38$ Evaluate the integral.
$$\int_{-1}^{2}(3 u-2)(u+1) d u$$

Calculus

James Stewart 8th Edition

Chapter 4, Problem 30

Instant Answer

Solved by Expert Anurag Kumar

03/18/2021

Step 1

The integral is a product of two binomials, so we can use the distributive property to simplify it: \begin{align*} \int_{-1}^{2}(3 u-2)(u+1) d u &= \int_{-1}^{2}(3u^2 + u - 2u - 2) du \\ &= \int_{-1}^{2}(3u^2 - u - 2) du. \end{align*} Show more…

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$19-38$ Evaluate the integral. $$\int_{-1}^{2}(3 u-2)(u+1) d u$$

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Key Concepts

Definite Integral

A definite integral represents the accumulation of quantities, such as areas under a curve, over a specific interval. It is evaluated by finding an antiderivative of the function and computing the difference of its values at the upper and lower limits of integration.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concepts of differentiation and integration, providing a method to evaluate definite integrals by using antiderivatives. It states that if a function is continuous on a closed interval, then its definite integral can be calculated using the difference between the antiderivative evaluated at the boundaries.

Polynomial Integration

Polynomial integration involves applying the power rule to integrate each term of a polynomial function individually. This technique simplifies the process of integrating complex polynomial expressions by breaking them down into their basic components.

Algebraic Expansion

Algebraic expansion is the process of multiplying out products of expressions to rewrite them as a sum of simpler terms. This simplification makes it easier to integrate the expression term by term, as it transforms products into sums that can be handled using standard integration techniques.

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