Chapter 5: Q. 91 (page 420)

Prove the integration formula
$\int \tan x d x = \ln | \sec x | + C$
(a) by using algebra and integration by substitution to find tan x dx;
(b) by differentiating ln | sec x|.

Short Answer

Expert verified

(a) After using algebra and integration by substitution we prove that $\int tanxdx = \ln (\sec) + C$

(b) After differentiating we prove that $\frac{d}{d x} \ln | \sec x | = \tan x$

Step by step solution

Step 1. Given Information

Prove the integration formula

$\int \tan x d x = \ln | \sec x | + C$

(a) by using algebra and integration by substitution to find $\int \tan x d x$ ;

(b) by differentiating $\ln | \sec x |$ .

Part (a) Step 1. Using algebra and integration by substitution to find ∫tanxdx.

We can write as

$\int \tan x d x = \int \frac{\sin x}{\cos x} d x$

Let

$u = \cos x \frac{d u}{d x} = - \sin x d u = - \sin x d x - d u = \sin x d x$

Part (a) Step 2. Now the integral after substitution.

$\int \tan x d x = - \int \frac{1}{u} du \int tanxdx = - \ln (u) + C \int tanxdx = - \ln (cosx) + C \int tanxdx = \ln (\sec x) + C$

Part (b) Step 1. By differentiating ln|secx|.

$\frac{d}{d x} \ln | \sec x | = \frac{d}{d x} \ln | \sec x | \cdot \frac{d}{d x} \sec x \frac{d}{d x} \ln | \sec x | = \frac{1}{\sec x} \cdot \sec x \tan x \frac{d}{d x} \ln | \sec x | = \tan x$

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Prove the integration formula
$\int \tan x d x = \ln | \sec x | + C$
(a) by using algebra and integration by substitution to find tan x dx;
(b) by differentiating ln | sec x|.

Short Answer

Step by step solution

Step 1. Given Information

Part (a) Step 1. Using algebra and integration by substitution to find ∫tanxdx.

Part (a) Step 2. Now the integral after substitution.

Part (b) Step 1. By differentiating ln|secx|.

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Prove the integration formula∫tanxdx=ln|secx|+C(a) by using algebra and integration by substitution to find tan x dx;(b) by differentiating ln | sec x|.

Short Answer

Step by step solution

Step 1. Given Information

Part (a) Step 1. Using algebra and integration by substitution to find ∫tanxdx.

Part (a) Step 2. Now the integral after substitution.

Part (b) Step 1. By differentiating ln|secx|.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Probability and Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Pure Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Prove the integration formula
$\int \tan x d x = \ln | \sec x | + C$
(a) by using algebra and integration by substitution to find tan x dx;
(b) by differentiating ln | sec x|.