Chapter 5: Q 83. (page 453)

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

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 $\int c s c x d x = - \ln | c s c x + c o t x | + C$

Part (a) Step 1. Explanation

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

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

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

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

Part (b) Step 1. Explanation

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

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

Short Answer

Step by step solution

Step 1. Given information

Part (a) Step 1. Explanation

Part (b) Step 1. Explanation

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

Short Answer

Step by step solution

Step 1. Given information

Part (a) Step 1. Explanation

Part (b) Step 1. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Discrete Mathematics

Mechanics Maths

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

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