What is the antiderivative of csc(x) cot(x)?
Understand the Problem
The question is asking for the antiderivative (also known as the indefinite integral) of the function csc(x) cot(x). This requires knowledge of integral calculus to solve it.
Answer
The antiderivative of $\csc(x) \cot(x)$ is $-\csc(x) + C$.
Answer for screen readers
The antiderivative of $\csc(x) \cot(x)$ is $-\csc(x) + C$.
Steps to Solve
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Identify the Integral to Solve
We need to find the antiderivative of the function $\csc(x) \cot(x)$. This can be represented as:
$$ \int \csc(x) \cot(x) , dx $$ -
Recall the Derivative
Recall that the derivative of $-\csc(x)$ is $\csc(x) \cot(x)$. Thus, the integral can be directly derived from this relationship. -
Write Down the Antiderivative
Using the derivative we identified, we can write the antiderivative:
$$ \int \csc(x) \cot(x) , dx = -\csc(x) + C $$
where $C$ is the constant of integration.
The antiderivative of $\csc(x) \cot(x)$ is $-\csc(x) + C$.
More Information
The antiderivative $\int \csc(x) \cot(x) , dx$ is commonly encountered in integral calculus. Knowing the derivatives of trigonometric functions can greatly simplify the process of finding antiderivatives.
Tips
- Forgetting to include the constant of integration $C$ when providing the final answer.
- Confusing the derivatives of trigonometric functions; it's key to remember which function corresponds to which derivative.
