Step 1: Understanding the Concept:
We are given an indefinite integral involving an exponential function and trigonometric functions. A very efficient strategy for this type of multiple-choice question is to differentiate the given options and see which one yields the original integrand.
Step 2: Key Formula or Approach:
By the Fundamental Theorem of Calculus, if $\int f(\theta) d\theta = F(\theta) + c$, then $F'(\theta) = f(\theta)$.
We use the Product Rule for differentiation: $\frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$, and the Chain Rule.
Step 3: Detailed Explanation:
Let's test Option (D): $F(\theta) = e^{\tan\theta} \cos\theta$
We differentiate $F(\theta)$ with respect to $\theta$:
\[ \frac{d}{d\theta} [e^{\tan\theta} \cos\theta] \]
Let $u = e^{\tan\theta}$ and $v = \cos\theta$.
The derivatives are:
\[ \frac{du}{d\theta} = e^{\tan\theta} \cdot \frac{d}{d\theta}(\tan\theta) = e^{\tan\theta} \sec^2\theta \]
\[ \frac{dv}{d\theta} = -\sin\theta \]
Applying the product rule:
\[ F'(\theta) = \left( e^{\tan\theta} \sec^2\theta \right) \cos\theta + e^{\tan\theta} (-\sin\theta) \]
Factor out $e^{\tan\theta}$:
\[ F'(\theta) = e^{\tan\theta} (\sec^2\theta \cos\theta - \sin\theta) \]
We know that $\sec\theta = \frac{1}{\cos\theta}$, so $\sec^2\theta \cos\theta = \sec\theta \cdot (\sec\theta \cos\theta) = \sec\theta \cdot 1 = \sec\theta$.
Substitute this back into the expression:
\[ F'(\theta) = e^{\tan\theta} (\sec\theta - \sin\theta) \]
This matches the original integrand perfectly. Thus, Option (D) is the correct antiderivative.
Step 4: Final Answer:
The correct value of the integral is $e^{\tan\theta} \cos\theta + c$. The correct option is (D).