What is the antiderivative of cosecant?

Steps to Solve

1. Rewrite $ ext{csc}(x)$ in terms of known integrals

   To make the integral easier to solve, rewrite $ ext{csc}(x)$ as $rac{1}{ ext{sin}(x)}$.

   $$ ext{csc}(x) = rac{1}{ ext{sin}(x)}$$

2. Multiply and divide by $ ext{csc}(x) + ext{cot}(x)$

   Multiply and divide the integrand by $ ext{csc}(x) + ext{cot}(x)$ to create a more solvable form.

   $$\int ext{csc}(x) , dx = \int \text{csc}(x) \cdot \frac{\text{csc}(x) + \text{cot}(x)}{\text{csc}(x) + \text{cot}(x)} , dx$$

3. Simplify the integral

   Simplify the integral to isolate a more straightforward derivative form

   $$\int \frac{\text{csc}^2(x) + \text{csc}(x) \text{cot}(x)}{\text{csc}(x) + \text{cot}(x)} , dx$$

4. Use substitution to solve the integral

   Let $u = \text{csc}(x) + \text{cot}(x)$, then calculate $du$.

   $$du = -(\text{csc}(x) \text{cot}(x) + \text{csc}^2(x)) , dx$$

   Hence,

   $$\int \frac{-1}{u} , du$$

5. Integrate and solve

   Integrate the function with respect to $u$.

   $$-\ln|u| + C$$ where $C$ is the constant of integration.

6. Substitute $u$ back in

   Replace $u$ with $\text{csc}(x) + \text{cot}(x)$.

   $$-\ln|\text{csc}(x) + \text{cot}(x)| + C$$