Question

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

• A. $e^{x}\tan x+C$
• B. $e^{x}\sec x+C$
• C. $e^{-x}\sec x+C$
• D. $e^{x}+\tan x+C$
• E. $e^{x}+\sec x+C$

Hint:

Integration Tip: Any time you see $e^x$ multiplied by a sum of two terms in an integral, immediately check if one term is the derivative of the other. It's the most common trick for $e^x$ integrals on exams!

Answer 1

The Correct Option is B

Concept:
There is a powerful standard integral identity involving the exponential function: $\int e^x [f(x) + f^{\prime}(x)] dx = e^x f(x) + C$. By distributing terms, we can format the given integral to perfectly match this identity.

Step 1: Distribute the terms inside the integral.
The given integral is: $$I = \int e^x \sec x(1 + \tan x) dx$$ Multiply $\sec x$ into the parentheses: $$I = \int e^x (\sec x + \sec x \tan x) dx$$

Step 2: Identify the function f(x).
Looking at the terms inside the bracket, set the first term as our primary function: $$f(x) = \sec x$$

Step 3: Find the derivative f'(x).
Differentiate $f(x)$ with respect to $x$ to see if it matches the second term: $$f^{\prime}(x) = \frac{d}{dx}(\sec x) = \sec x \tan x$$ This perfectly matches the second term in our integral.

Step 4: Recognize the standard integral form.
The integral is now verified to be in the exact format of the identity: $$I = \int e^x [f(x) + f^{\prime}(x)] dx$$

Step 5: Apply the theorem for the final answer.
Using the identity, the evaluated integral is simply $e^x f(x) + C$: $$I = e^x \sec x + C$$ Hence the correct answer is (B) $e^{x\sec x+C$}.