Question

Evaluate the integral: \[ \int e^x \sec(x) \left( \tan(x) + 1 \right) \, dx \]

• A. \( e^x \sec(x) + C \)
• B. \( e^x \sec(x) \left( \tan(x) + 1 \right) + C \)
• C. \( e^x \sec(x) \tan(x) + C \)
• D. \( e^x \sec(x) \tan(x) + e^x \sec(x) + C \)

Hint:

When encountering integrals involving secant and tangent functions, look for opportunities to apply known derivatives such as \( \frac{d}{dx} \sec(x) = \sec(x) \tan(x) \) to simplify the integral.

Answer 1

The Correct Option is D

Evaluate the integral: \[ I = \int e^x \sec(x) \left( \tan(x) + 1 \right) \, dx \]

1. Step 1: Simplify the integrand: The integrand can be simplified by expanding the terms inside the brackets: \[ \sec(x) \left( \tan(x) + 1 \right) = \sec(x) \tan(x) + \sec(x) \] Thus, the integral becomes: \[ I = \int e^x \left( \sec(x) \tan(x) + \sec(x) \right) \, dx \]

2. Step 2: Separate the terms: We can now split the integral into two parts: \[ I = \int e^x \sec(x) \tan(x) \, dx + \int e^x \sec(x) \, dx \]

3. Step 3: Integrate each part: - The first integral \( \int e^x \sec(x) \tan(x) \, dx \) can be solved by recognizing that the derivative of \( \sec(x) \) is \( \sec(x) \tan(x) \), so this part integrates to: \[ e^x \sec(x) + C_1 \] - The second integral \( \int e^x \sec(x) \, dx \) can be integrated similarly, resulting in: \[ e^x \sec(x) + C_2 \]

4. Step 4: Combine the results: Adding the two integrals gives us the final result: \[ I = e^x \sec(x) \tan(x) + e^x \sec(x) + C \]