Chapter 5: Q 82. (page 453)

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

Short Answer

Expert verified

(a) Formula is proved by multiplying the integrand by a form of 1

(b) Formula is proved by differentiating ln | sec x + tan x|.

Step by step solution

Part (a) Step 1. Explanation

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

Let width="117">secx+tanx=u

$s e c x \tan x + s e c^{2} x d x = d u$

$\begin{array}{rcl} 鈭� s e c x d x & = & 鈭� \frac{d u}{u} \\ = & \ln u + C \\ = & \ln (s e c x + \tan x) + C \end{array}$

Part (b) Step 1. Explanation

$\begin{array}{rcl} \frac{d}{d x} (\ln |s e c x + \tan x|) & = & \frac{s e c x \tan x + s e c^{2} x}{|s e c x + \tan x|} \\ = & \frac{s e c x (\tan x + s e c x)}{s e c x + \tan x} \\ = & s e c x \\ 鈬� & 鈭� s e c x = \ln |s e c x + \tan x| + C \end{array}$

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91影视

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

Short Answer

Step by step solution

Part (a) Step 1. Explanation

Part (b) Step 1. Explanation

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91影视

Prove the integration formula 鈭�secxdx=ln|secx+tanx|+C(a) by multiplying the integrand by a form of 1 and then applying a substitution;(b) by differentiating ln | sec x + tan x|.

Short Answer

Step by step solution

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

Pure Maths

Statistics

Calculus

Applied Mathematics

Discrete Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

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