Chapter 5: Q 83. (page 453)

Prove the integration formula $鈭� c s c x d x = 鈭� \ln | c s c x + c o t x | + C$
(a) by multiplying the integrand by a form of 1 and then applying a substitution;
(b) by differentiating $鈭� \ln | c s c x + c o t x |$ .

Short Answer

Expert verified

(a) Formula is proved by multiplying the integrand by a form of 1 and then applying a substitution.

(b) Formula is proved by differentiating.

Step by step solution

Step 1. Given information

An integral is $鈭� c s c x d x = 鈭� \ln | c s c x + c o t x | + C$

Part (a) Step 1. Explanation

$\begin{array}{rcl} 鈭� c s c x d x & = & 鈭� \frac{c s c x (c s c x + c o t x)}{c s c x + c o t x} d x \\ = & 鈭� \frac{c s c^{2} x + c s c x c o t x}{c s c x + c o t x} d x \end{array}$

Let $c s c x + c o t x = u$

width="205">-cscxcotx-csc2xdx=du

$\begin{array}{rcl} 鈭� c s c x d x & = & 鈭� - d u \\ = & - u + C \\ = & - c s c x - c o t x + C \end{array}$

Part (b) Step 1. Explanation

$\begin{array}{rcl} \frac{d}{d x} [- \ln (c s c x + c o t x)] & = & \frac{- 1 (- c s c^{2} x - c s c x c o t x)}{c s c x + c o t x} \\ = & c s c x \\ 鈬� & 鈭� c s c x d x = - \ln (c s c x + c o t x) + C \end{array}$

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91影视

Prove the integration formula $鈭� c s c x d x = 鈭� \ln | c s c x + c o t x | + C$
(a) by multiplying the integrand by a form of 1 and then applying a substitution;
(b) by differentiating $鈭� \ln | c s c x + c o t x |$ .

Short Answer

Step by step solution

Step 1. Given information

Part (a) Step 1. Explanation

Part (b) Step 1. Explanation

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91影视

Prove the integration formula 鈭�cscxdx=鈭�ln|cscx+cotx|+C(a) by multiplying the integrand by a form of 1 and then applying a substitution;(b) by differentiating 鈭�ln|cscx+cotx|.

Short Answer

Step by step solution

Step 1. Given information

Part (a) Step 1. Explanation

Part (b) Step 1. Explanation

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

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Study anywhere. Anytime. Across all devices.

Prove the integration formula $鈭� c s c x d x = 鈭� \ln | c s c x + c o t x | + C$
(a) by multiplying the integrand by a form of 1 and then applying a substitution;
(b) by differentiating $鈭� \ln | c s c x + c o t x |$ .