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Q. 52

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Calculus

Found in: Page 353

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Calculate the exact value of each definite integral in Exercises 47–52 by using properties of definite integrals and the formulas in Theorem 4.13.
$\int_{6}^{1} (3 (1 - 2 x)^{2} + 4 x) d x$

The exact value of definite integral is $- 735$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information

We are given,

$\int_{6}^{1} (3 (1 - 2 x)^{2} + 4 x) d x$

Step 2. Finding the Integral

The definite integral is given by,

$\int_{6}^{1} (3 (1 - 2 x)^{2} + 4 x) d x = \int_{6}^{1} (3 (1 - 4 x + 4 x^{2}) + 4 x) d x = \int_{6}^{1} (3 - 12 x + 12 x^{2} + 4 x) d x = 3 \int_{6}^{1} 1 d x - 8 \int_{6}^{1} x d x + 12 \int_{6}^{1} x^{2} d x = 3 [1 - 6] - 8 [\frac{1}{2} (1 - 36)] + 12 [\frac{1}{3} (1 - 216)] = - 15 - 4 (- 35) + 4 (- 215) = - 15 + 140 - 860 = - 735$

Most popular questions for Math Textbooks

If f is negative on [−3, 2], is the definite integral $\int_{- 3}^{2} f (x) d x$ positive or negative? What about the definite integral − $\int_{- 3}^{2} f (x) d x$ ?

Write out all the integration formulas and rules that we know at this point.

Fill in each of the blanks:

(a) $\int x^{6} d x = + C .$

(b) is an antiderivative of role="math" localid="1648619282178" $x^{6} .$

(c) The derivative of is $x^{6} .$

Properties of addition: State the associative law for addition, the commutative law for addition, and the distributive law for multiplication over addition of real numbers. (You may have to think back to a previous algebra course.)

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

$\int_{- 3}^{2} \frac{1}{x + 5} d x$

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