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Q. 14

Expert-verified

Found in: Page 362

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Verify that $\int \cot x d x = \ln (\sin x) + C$ . (Do not try to solve the integral from scratch.)

It is verified that $\int \cot x d x = \ln (\sin x) + C .$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

The given integral is $\int \cot x d x = \ln (\sin x) + C .$

Step 2. Verification.

We can write the differentiation as,

$\frac{d}{d x} \ln (\sin x) = \frac{1}{\sin x} (\cos x)$

$= \frac{\cos x}{\sin x} = c o t x$

Hence, $\ln (\sin x) is an anti-derivative of \cot x .$

Therefore,

$\int \cot x d x = \ln (\sin x) + C$

Most popular questions for Math Textbooks

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The absolute area between the graph of f and the x-axis on [a, b] is equal to $| \int_{a}^{b} f (x) d x |$ .

(b) True or False: The area of the region between f(x) = x − 4 and g(x) = $- x^{2}$ on the interval [−3, 3] is negative.

(c) True or False: The signed area between the graph of f on [a, b] is always less than or equal to the absolute area on the same interval.

(d) True or False: The area between any two graphs f and g on an interval [a, b] is given by $\int_{a}^{b} (f (x) - g (x)) d x$ .

(e) True or False: The average value of the function f(x) = $x^{2} - 3$ on [2, 6] is

\frac{f (6) + f (2)}{2}

=

\frac{33 + 1}{2}

= 17.

(f) True or False: The average value of the function f(x) = $x^{2} - 3$ on [2, 6] is $\frac{f (6) - f (2)}{4}$ = $\frac{33 - 1}{4}$ = 8.

(g) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 2] and the average value of f on [2, 5].

(h) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 3] and the average value of f on [3, 5].

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.

$\int_{5}^{2} (9 + 10 x - x^{2}) d x$

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

$\int_{0}^{1} \frac{1}{1 + x^{2}} d x$

Use the graph of f to estimate the values of A(1), A(2), A(3)

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.

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

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Calculus

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Short Answer

Verify that $\int \cot x d x = \ln (\sin x) + C$ . (Do not try to solve the integral from scratch.)

Step by Step Solution

Step 1. Given information.

Step 2. Verification.

Most popular questions for Math Textbooks

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Calculus

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Short Answer

Verify that∫cotxdx=ln(sinx)+C. (Do not try to solve the integral from scratch.)

Step by Step Solution

Step 1. Given information.

Step 2. Verification.

Most popular questions for Math Textbooks

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Verify that $\int \cot x d x = \ln (\sin x) + C$ . (Do not try to solve the integral from scratch.)