What is the integral of csc(x) cot(x)?
Understand the Problem
The question is asking for the integral of the function csc(x) cot(x). This involves finding the antiderivative of this trigonometric expression, which can be solved using standard integrals.
Answer
$$ -csc(x) + C $$
Answer for screen readers
The integral of $csc(x) \cdot cot(x)$ is:
$$ -csc(x) + C $$
Steps to Solve
- Identify the integral to solve
We need to find the integral of the function $csc(x) \cdot cot(x)$, which can be represented as:
$$ \int csc(x) \cdot cot(x) , dx $$
- Rewrite the integrand
Recall that the function $csc(x)$ can be expressed as $\frac{1}{sin(x)}$ and $cot(x)$ as $\frac{cos(x)}{sin(x)}$. Therefore, we can rewrite the integral as:
$$ \int \frac{cos(x)}{sin^2(x)} , dx $$
- Use a substitution
Let $u = sin(x)$. Then, the differential $du$ becomes $cos(x) , dx$. Thus, we rewrite the integral in terms of $u$:
$$ \int \frac{1}{u^2} , du $$
- Integrate using power rule
Integrating $\frac{1}{u^2}$ gives us:
$$ -\frac{1}{u} + C $$
where $C$ is the constant of integration.
- Substitute back to original variable
Now substitute back $u = sin(x)$:
$$ -\frac{1}{sin(x)} + C $$
- Express in terms of $csc(x)$
Recalling that $csc(x) = \frac{1}{sin(x)}$, we can write:
$$ -csc(x) + C $$
The integral of $csc(x) \cdot cot(x)$ is:
$$ -csc(x) + C $$
More Information
The result shows that the integral of the product of the cosecant and cotangent functions leads to a negative cosecant function plus a constant. This type of integral is commonly found in calculus, particularly in problems involving trigonometric functions.
Tips
- Confusing $cot(x)$ and $csc(x)$ during substitution.
- Forgetting to change the limits of integration if the integral were definite (though this is not the case here).
- Not recognizing the need to substitute back to the original variable after integrating.
