Integration of cot x
Understand the Problem
The question is asking for the integral of the function cot(x), which is a trigonometric function. To solve it, we will need to apply integration techniques and recognize the relationship of cot(x) with other trigonometric functions.
Answer
\ln| \sin(x) | + C
Answer for screen readers
The final answer is $\int \cot(x) , dx = \ln| \sin(x) | + C$
Steps to Solve
(! = step title followed by the explanation that follows it using latex at right time, in correct order) ! Understand the function cot(x) as a ratio function Given the function to integrate, cot(x), rewrite it in terms of sin(x) and cos(x): [ \cot(x) = \frac{\cos(x)}{\sin(x)} ] ! Recognize the integral form Recognize that the integral of cot(x) is a standard integral. We can use the identity of the integral: [ \int \cot(x) , dx = \int \frac{\cos(x)}{\sin(x)} , dx ] ! Use u-substitution for integration Use a substitution where we set ( u = \sin(x) ), hence ( du = \cos(x) , dx ): [ \int \cot(x) , dx = \int \frac{\cos(x)}{\sin(x)} , dx = \int \frac{1}{u} , du ] ! Solve the integral of 1/u We know that ( \int \frac{1}{u} , du = \ln|u| + C ), so: [ \int \cot(x) , dx = \ln| \sin(x) | + C ]
The final answer is $\int \cot(x) , dx = \ln| \sin(x) | + C$
More Information
Cotangent is a common trigonometric function, and knowing its integral is very useful in both math and physics contexts.
Tips
A common mistake is forgetting the absolute value sign in the logarithm function when integrating 1/u.
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