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Chapter 5: Q. 92 (page 420)

Prove the integration formula

鈭�cotxdx=鈭�ln|cscx|+C

(a) by using algebra and integration by substitution to find 鈭�cotxdx;

(b) by differentiating 鈭�ln|cscx|.

Short Answer

Expert verified

(a) After using algebra and integration by substitution鈭�cotxdx=-lncscx+C.

(b) After differentiating鈭�ddxln|cscx|=cotx.

Step by step solution

01

Step 1. Given Information 

Prove the integration formula

鈭�cotxdx=鈭�ln|cscx|+C

(a) by using algebra and integration by substitution to find 鈭�cotxdx;

(b) by differentiating 鈭�ln|cscx|.

02

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

We can write as

鈭�cotxdx=鈭�cosxsinxdx

Let

u=sinxdudx=cosxdu=cosxdx

03

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

鈭�cotxdx=鈭�1udu鈭�cotxdx=lnu+C鈭�cotxdx=lnsinx+C鈭�cotxdx=-lncscx+C

04

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

鈭�ddxln|cscx|=鈭�ddxln|cscx|路ddxcscx鈭�ddxln|cscx|=鈭�1cscx路(-cscxcotx)鈭�ddxln|cscx|=cotx

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