Chapter 5: Q. 92 (page 420)

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

Short Answer

Expert verified

(a) After using algebra and integration by substitution $\int \cot x d x = - \ln (cscx) + C$ .

(b) After differentiating $- \frac{d}{d x} \ln | \csc x | = \cot x$ .

Step by step solution

Step 1. Given Information

Prove the integration formula

$\int \cot x d x = - \ln | \csc x | + C$

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

(b) by differentiating $- \ln | \csc x |$ .

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

We can write as

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

Let

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

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

$\int \cot x d x = \int \frac{1}{u} du \int \cot x d x = \ln (u) + C \int \cot x d x = \ln (\sin x) + C \int \cot x d x = - \ln (cscx) + C$

Part (b) Step 1. By differentiating −ln|cscx|

$- \frac{d}{d x} \ln | \csc x | = - \frac{d}{d x} \ln | \csc x | \cdot \frac{d}{d x} \csc x - \frac{d}{d x} \ln | \csc x | = - \frac{1}{\csc x} \cdot (- \csc x \cot x) - \frac{d}{d x} \ln | \csc x | = \cot x$

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

Short Answer

Step by step solution

Step 1. Given Information

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

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

Part (b) Step 1. By differentiating −ln|cscx|

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Prove the integration formula∫cotxdx=−ln|cscx|+C(a) by using algebra and integration by substitution to find ∫cotxdx;(b) by differentiating −ln|cscx|.

Short Answer

Step by step solution

Step 1. Given Information

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

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

Part (b) Step 1. By differentiating −ln|cscx|

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Decision Maths

Geometry

Calculus

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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