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Chapter 4: Q. 51 (page 353)

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.
${鈭�}_{0}^{4} ((2 x - 3)^{2} + 5) d x$

Short Answer

Expert verified

The exact value of definite integral is $\frac{136}{3}$ .

Step by step solution

01

Step 1. Given Information

We are given,

${鈭�}_{0}^{4} ((2 x - 3)^{2} + 5) d x$

02

Step 2. Finding the Integral

The definite integral is given by,

${鈭�}_{0}^{4} ((2 x - 3)^{2} + 5) d x = {鈭�}_{0}^{4} (4 x^{2} - 12 x + 9 + 5) d x = {鈭�}_{0}^{4} (4 x^{2} - 12 x + 14) d x = 4 {鈭�}_{0}^{4} x^{2} d x - 12 {鈭�}_{0}^{4} x d x + {鈭�}_{0}^{4} 14 d x = 4 (\frac{1}{3} (4^{3})) - 12 (\frac{1}{2}) (4^{2}) + 14 (4) = 4 (\frac{64}{3}) - 6 (16) + 56 = \frac{256}{3} - 96 + 56 = \frac{256 - 288 + 168}{3} = \frac{136}{3}$

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Most popular questions from this chapter

Given a simple proof that ${鈭�}_{k = 5}^{n} (a_{k} + b_{k}) = {鈭�}_{k = 5}^{n} a_{k} + {鈭�}_{k = 5}^{n} b_{k}$

What is the difference between an antiderivative of a function and the indefinite integral of a function?

If ${鈭�}_{- 2}^{3} f (x) d x = 4, {鈭�}_{- 2}^{6} f (x) d x = 9, {鈭�}_{- 2}^{3} g (x) d x = 2$ and ${鈭�}_{3}^{6} g (x) d x = 3$ ,then find the values of each definite integral in Exercises $29 - 40$ . If there is not enough information, explain why.

${鈭�}_{3}^{6} f (x) d x$ .

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value

$\lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} \frac{{(k + 1)}^{2}}{n^{3} - 1}$

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value

$\lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} \frac{k^{2} + k + 1}{n^{2}}$

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