To solve the integral \( \int \sec(x) \, dx \), we use a standard technique involving multiplying by a clever form of 1, which is:

\[
\int \sec(x) \, dx = \int \frac{\sec(x)(\sec(x) + \tan(x))}{\sec(x) + \tan(x)} \, dx
\]

Now, let’s proceed with the steps:

### Step 1: Multiply by the expression
\[
\int \frac{\sec(x)(\sec(x) + \tan(x))}{\sec(x) + \tan(x)} \, dx = \int \frac{d(\sec(x) + \tan(x))}{\sec(x) + \tan(x)}
\]
This works because \( d(\sec(x) + \tan(x)) = \sec(x)\tan(x) + \sec^2(x) \, dx \), which simplifies the problem.

### Step 2: Substitute
Let:
\[
u = \sec(x) + \tan(x)
\]
Then:
\[
du = (\sec(x)\tan(x) + \sec^2(x)) \, dx
\]
Thus, the integral becomes:
\[
\int \frac{du}{u} = \ln|u| + C
\]

### Step 3: Substitute back
Substitute \( u = \sec(x) + \tan(x) \) back into the result:
\[
\int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C
\]

So the solution is:
\[
\boxed{\ln|\sec(x) + \tan(x)| + C}
\]

Would you like more details or have any further questions?

Here are 5 related questions to explore:
1. How would you compute \( \int \sec^2(x) \, dx \)?
2. What is the derivative of \( \ln|\sec(x) + \tan(x)| \)?
3. Can you compute \( \int \sec^3(x) \, dx \)?
4. What is the integral of \( \int \csc(x) \, dx \)?
5. How does the substitution method help simplify integrals involving trigonometric functions?

**Tip:** When dealing with integrals of trigonometric functions, try to recall known integrals and useful substitutions to make the process faster.

Learn how to solve the integral ∫ sec(x) dx using the substitution method. This problem covers key trigonometric and integration concepts, including how to apply a clever substitution to simplify the integral. Suitable for advanced high school students or college-level learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation