Understanding the Integral of Cotangent

The video tutorial explains how to compute the integral of 7 cotangent x dx using two methods. The first method involves direct integration using a known formula, resulting in 7 times the natural log of the absolute value of sin x plus a constant. The second method uses substitution, where cotangent x is expressed as cosine x divided by sin x, and substitution is performed with u equal to sin x. This method also leads to the same result, providing clarity on why the natural log function appears in the antiderivative of the cotangent function.

What is the result of integrating 7 cotangent x dx using the standard formula?

• 7 times the natural log of the absolute value of cot x plus c

• 7 times the natural log of the absolute value of tan x plus c

• 7 times the natural log of the absolute value of cos x plus c

• 7 times the natural log of the absolute value of sin x plus c

In the alternative method, what substitution is made to solve the integral of cotangent x?

• u = cot x

• u = cos x

• u = sin x

• u = tan x

What is the integral of 7 times cosine x divided by sin x dx after substitution?

• 7 times the natural log of the absolute value of sin x plus c

• 7 times the natural log of the absolute value of tan x plus c

• 7 times the natural log of the absolute value of cos x plus c

• 7 times the natural log of the absolute value of cot x plus c

Why is the natural log function used in the antiderivative of the cotangent function?

• Because it simplifies the expression

• Because it is a standard result of integrating trigonometric functions

• Because it appears naturally when using substitution

• Because it is the only function that fits the integral

What is the purpose of using a second method to find the antiderivative of cotangent x?

• To verify the result using a different approach

• To make the process more complex

• To find a completely different result

• To avoid using the natural log function