Q. 19 Are definite integrals the 鈥渋n... [FREE SOLUTION]

Chapter 4: Q. 19 (page 399)

Are definite integrals the 鈥渋nverse鈥� of differentiation? In other words, does one undo the other? Simplify each of the following to answer this question:
$(a) {鈭�}_{a}^{b} f^{'} (x) d x (b) \frac{d}{d x} {鈭�}_{a}^{b} f (x) d x$

Short Answer

Expert verified

Part $(a)$ The simplified expression for ${鈭�}_{a}^{b} f^{'} (x) d x$ is, $f (b) - f (a)$ .

Part $(b)$ The simplified expression forrole="math" localid="1648811812391" $\frac{d}{d x} {鈭�}_{a}^{b} f (x) d x$ is, $0$ .

Step by step solution

Part a Step 1. Given information

${鈭�}_{a}^{b} f^{'} (x) d x$ .

Part a Step 2. The objective is, we need to simplify the given expression.

$\frac{d}{d x} f (x) = f^{'} (x)$

So,

$鈭� f^{'} (x) d x = [f (x)]_{a}^{b} = f (b) - f (a)$

Part b Step 1. Given information

$\frac{d}{d x} {鈭�}_{a}^{b} f (x) d x$ .

Part b Step 2. The objective is, we need to simplify the given expression.

$F (x) = f (x)$ [ $F (x)$ is an antiderivative of $f (x)$ ]

So,

$\frac{d}{d x} {鈭�}_{a}^{b} f (x) d x = \frac{d}{d x} [F (x)]_{a}^{b} = \frac{d}{d x} [F (b) - F (a)]$

$= 0$ [ $F (b) - F (a)$ is a constant and derivative of a constant is $0$ ]

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Are definite integrals the 鈥渋nverse鈥� of differentiation? In other words, does one undo the other? Simplify each of the following to answer this question:
$(a) {鈭�}_{a}^{b} f^{'} (x) d x (b) \frac{d}{d x} {鈭�}_{a}^{b} f (x) d x$

Short Answer

Step by step solution

Part a Step 1. Given information

Part a Step 2. The objective is, we need to simplify the given expression.

Part b Step 1. Given information

Part b Step 2. The objective is, we need to simplify the given expression.

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91影视

Are definite integrals the 鈥渋nverse鈥� of differentiation? In other words, does one undo the other? Simplify each of the following to answer this question:(a)鈭�abf'(x)dx(b)ddx鈭�abf(x)dx

Short Answer

Step by step solution

Part a Step 1. Given information

Part a Step 2. The objective is, we need to simplify the given expression.

Part b Step 1. Given information

Part b Step 2. The objective is, we need to simplify the given expression.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Statistics

Geometry

Mechanics Maths

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Are definite integrals the 鈥渋nverse鈥� of differentiation? In other words, does one undo the other? Simplify each of the following to answer this question:
$(a) {鈭�}_{a}^{b} f^{'} (x) d x (b) \frac{d}{d x} {鈭�}_{a}^{b} f (x) d x$