Integral of csc cot
Understand the Problem
The question is asking for the integral of the function csc(x) * cot(x). This requires us to find the antiderivative of this trigonometric expression.
Answer
The integral of $ \csc(x) \cot(x) $ is $$ -\csc(x) + C $$
Answer for screen readers
The integral of $ \csc(x) \cot(x) $ is
$$ -\csc(x) + C $$
Steps to Solve
- Set up the integral
We need to find the integral of the function $ \csc(x) \cot(x) $. This can be expressed as:
$$ \int \csc(x) \cot(x) , dx $$
- Identify a useful substitution
Notice that the derivative of $ \csc(x) $ is $ -\csc(x) \cot(x) $. This suggests that $ \csc(x) $ is a suitable substitution since its derivative is related to the integrand.
Let: $$ u = \csc(x) $$ Then, the derivative is: $$ \frac{du}{dx} = -\csc(x) \cot(x) $$ or rearranging gives: $$ du = -\csc(x) \cot(x) , dx $$
- Rewrite the integral in terms of u
We can rewrite the integral using our substitution. First, we can write $ \csc(x) \cot(x) $ as $ -du $:
$$ \int \csc(x) \cot(x) , dx = -\int du $$
- Integrate
The integral of $ -1 $ with respect to $ u $ is simply: $$ -u + C $$
- Substitute back
Now, we substitute back $ u = \csc(x) $ to obtain the final result:
$$ -\csc(x) + C $$
The integral of $ \csc(x) \cot(x) $ is
$$ -\csc(x) + C $$
More Information
The integrals of trigonometric functions sometimes involve recognizing patterns and relationships between functions and their derivatives. Here, recognizing the relationship between the derivative of $ \csc(x) $ and the integrand was crucial for solving the problem.
Tips
- Forgetting to account for the negative sign from the substitution can lead to ( \csc(x) + C ) instead of ( -\csc(x) + C ). It’s important to track sign changes carefully when using substitutions.
