Question

Match the pairs correctly: 

(i)  \( \int \tan x \, dx \)  \(\hspace{3.5cm}\)  \( \log |\sin x| + c \) 
(ii)  \( \int \cot x \, dx \)  \(\hspace{3.5cm}\)  \( \log |\csc x| - \cot x + c \) 
(iii)  \( \int \sec x \, dx \)  \(\hspace{3.5cm}\)  \( \log |\sec x + \tan x| + c \) 
(iv)  \( \int \csc x \, dx \)  \(\hspace{3.5cm}\)  \( -\log |\csc x + \cot x| + c \) 
(v)  \( \int \frac{\cos x}{\sin x} \, dx \)  \(\hspace{3.5cm}\)  \( \log |\sin x| + c \) 
(vi)  Derivative of \( \sin 2x \) with respect to \( x \)  \(\hspace{0.75cm}\)  \( 2 \cos 2x \) 

Hint:

When matching integrals and derivatives, look for standard identities and formulas to simplify the expressions.

Answer 1

Step 1: Understanding the integrals and derivatives.
- (i) The integral of \( \tan x \) is \( \log |\sin x| + c \) based on the standard integral formula for \( \tan x \). - (ii) The integral of \( \cot x \) is \( \log |\csc x| - \cot x + c \). - (iii) The integral of \( \sec x \) is \( \log |\sec x + \tan x| + c \). - (iv) The integral of \( \csc x \) is \( -\log |\csc x + \cot x| + c \). - (v) The integral of \( \frac{\cos x}{\sin x} \) is \( \log |\sin x| + c \), as this is the standard form of the cotangent integral. - (vi) The derivative of \( \sin 2x \) is \( 2 \cos 2x \), using the chain rule for the derivative of a sine function.

Step 2: Conclusion.
The correct matches are: \[ (i) \, \text{log} |\sin x| + c, (ii) \, \log |\csc x| - \cot x + c, (iii) \, \log |\sec x + \tan x| + c, (iv) \, -\log |\csc x + \cot x| + c, \] \[ (v) \, \log |\sin x| + c, (vi) \, 2 \cos 2x. \]