Integral of csc squared
Understand the Problem
The question is asking for the integral of the function csc squared, which is a trigonometric function. To solve it, we will use the knowledge of integration techniques in calculus to determine the antiderivative.
Answer
$$ -\cot(x) + C $$
Answer for screen readers
$$ -\cot(x) + C $$
Steps to Solve
- Recognize the function to integrate
We want to find the integral of the function $\csc^2(x)$. The integral can be expressed as:
$$ \int \csc^2(x) , dx $$
- Recall the antiderivative
The antiderivative of $\csc^2(x)$ is a well-known result. It can be computed, and we find that:
$$ \int \csc^2(x) , dx = -\cot(x) + C $$
where $C$ is the constant of integration.
- Write the final answer
Combining the previous steps, the final solution to the integral is:
$$ -\cot(x) + C $$
$$ -\cot(x) + C $$
More Information
The integral of $\csc^2(x)$ results in the negative cotangent function, $-\cot(x)$, plus a constant of integration, $C$. This relationship is a fundamental result in trigonometric integrals commonly used in calculus.
Tips
- Forgetting the constant of integration: Always include $C$ when finding an indefinite integral.
- Confusing with other trigonometric functions: Make sure to remember that $\int \csc^2(x) , dx$ specifically leads to $-\cot(x)$, and not to other functions like $\sec^2(x)$ or $-\csc(x)$.
