Question

$$ \int e^x (\tan x + \sec^2 x) \, dx = ? $$

• A. $e^x \sec x + c$
• B. $e^x \tan x + c$
• C. $e^x \cot x + c$
• D. $e^x \tan^2 x + c$

Hint:

Always check the derivative of the first term inside the bracket. In trigonometric forms involving $e^x$, common pairs are $(\sin x, \cos x)$, $(\tan x, \sec^2 x)$, and $(\sec x, \sec x \tan x)$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Similar to the previous question, this follows the standard identity $\int e^x [f(x) + f'(x)] \, dx = e^x f(x) + c$.

Step 2: Key Formula or Approach:
Identify $f(x)$ and check if the second term in the bracket is $f'(x)$.

Step 3: Detailed Explanation:
1. Let $f(x) = \tan x$. 2. Differentiate $f(x)$ with respect to $x$: \[ f'(x) = \frac{d}{dx}(\tan x) = \sec^2 x \] 3. Observe that the integrand is exactly $e^x [f(x) + f'(x)]$. 4. Use the identity: \[ \int e^x (\tan x + \sec^2 x) \, dx = e^x f(x) + c = e^x \tan x + c \]

Step 4: Final Answer
The result of the integral is $e^x \tan x + c$.