Chapter 5: Q. 5. (page 451)

Given the algebraic trick for integrating $s e c x$ used in the proof of Theorem 5.17, what do you think is the algebraic trick used for integrating $c s c x$ ?

Short Answer

Expert verified

Multiply the integrand by $\frac{c s c x + c o t x}{c s c x + c o t x}$ .

Step by step solution

Step 1. Given Information.

Neither integration by substitution nor integration by parts can immediately improve the integral $\int s e c x d x$ . However, there is a particular technique that happens to apply to this integral. If we multiply the integrand by a certain very special form of 1, we will have an integral that can be evaluated by substitution: $s e c x + \tan x d x$ .

Step 2. Method

We can calculate the integral of $\int c s c x d x$ by multiplying the integrand in both numerator and denominator by $\frac{c s c x + c o t x}{c s c x + c o t x}$ .

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Given the algebraic trick for integrating $s e c x$ used in the proof of Theorem 5.17, what do you think is the algebraic trick used for integrating $c s c x$ ?

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Method

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Given the algebraic trick for integrating secxused in the proof of Theorem 5.17, what do you think is the algebraic trick used for integrating cscx?

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Method

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Given the algebraic trick for integrating $s e c x$ used in the proof of Theorem 5.17, what do you think is the algebraic trick used for integrating $c s c x$ ?