integrate sec x

Steps to Solve

1. Rewrite the integral using a known technique

   We start with the integral of $ ext{sec}(x)$, and use a clever substitution technique to simplify it. Specifically, we multiply and divide by $ ext{sec}(x) + ext{tan}(x)$.

   $$ \int \text{sec}(x) , dx $$

   Multiply and divide by $\text{sec}(x) + \text{tan}(x)$:

   $$ \int \frac{\text{sec}(x)(\text{sec}(x) + \text{tan}(x))}{\text{sec}(x) + \text{tan}(x)} , dx $$

2. Simplify the integrand

   Simplify the expression inside the integral:

   $$ \int \frac{\text{sec}^2(x) + \text{sec}(x)\text{tan}(x)}{\text{sec}(x) + \text{tan}(x)} , dx $$

   Let $u = \text{sec}(x) + \text{tan}(x)$. Then, the differential $du$ is:

   $$ du = \text{sec}(x)\text{tan}(x) , dx + \text{sec}^2(x) , dx $$

   Thus,

   $$ du = (\text{sec}(x)\text{tan}(x) + \text{sec}^2(x)) , dx $$

3. Perform the substitution

   Notice that the numerator of our integral $\text{sec}^2(x) + \text{sec}(x)\text{tan}(x)$ is exactly $du$:

   $$ \int \frac{du}{u} $$

4. Integrate using the natural logarithm rule

   The integral of $\frac{1}{u}$ with respect to $u$ is $\ln|u| + C$:

   $$ \ln|u| + C $$

5. Substitute back the original expression

   Substitute $u = \text{sec}(x) + \text{tan}(x)$ back into the expression:

   $$ \ln|\text{sec}(x) + \text{tan}(x)| + C $$