Question

\( \int e^x \sec x (1+\tan x) dx = \)

• A. \( e^x \sec^2 x + C \)
• B. \( e^x \tan x + C \)
• C. \( e^x \sec x + C \)
• D. \( e^x \tan^2 x + C \)
• E. \( e^x \sec x \tan x + C \)

Hint:

Always check if integrand matches derivative of a product.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This integral is a specific application of a standard integration formula involving the exponential function \(e^x\). The formula simplifies the integration of \(e^x\) multiplied by the sum of a function and its derivative.
Step 2: Key Formula or Approach:
The key formula is: \[ \int e^x [f(x) + f'(x)] dx = e^x f(x) + C \] This formula is derived from the product rule for differentiation. The derivative of \(e^x f(x)\) is \(e^x f(x) + e^x f'(x) = e^x [f(x) + f'(x)]\). Our goal is to get the integrand into the form \(e^x [f(x) + f'(x)]\).
Step 3: Detailed Explanation:
First, distribute \(\sec x\) inside the parenthesis: \[ \int e^x (\sec x + \sec x \tan x) dx \] Now, we need to check if this expression fits the form \(\int e^x [f(x) + f'(x)] dx\). Let's try setting \(f(x) = \sec x\). Now, we find the derivative of \(f(x)\): \[ f'(x) = \frac{d}{dx}(\sec x) = \sec x \tan x \] This matches the second term in the parenthesis. So, our integral is exactly in the standard form with \(f(x) = \sec x\) and \(f'(x) = \sec x \tan x\). Using the formula: \[ \int e^x [\sec x + \sec x \tan x] dx = e^x \sec x + C \] Step 4: Final Answer:
The integral is \(e^x \sec x + C\).