Question

$\int e^{2\theta}(2\cos^{2}\theta-\sin 2\theta)\, d\theta =$

• A. $e^{2\theta}\cos^{2}\theta+C$
• B. $e^{2\theta}\sin 2\theta+C$
• C. $2e^{2\theta}\cos^{2}\theta+C$
• D. $e^{2\theta}\sin \theta+C$
• E. $e^{2\theta}\cos 2\theta+C$

Hint:

$\int e^x (f(x) + f'(x)) dx = e^x f(x) + C$ is a very common identity in entrance exams.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We are asked to integrate a function which is a product of an exponential term and a trigonometric term. This structure suggests a special integration formula related to the product rule of differentiation.
Step 2: Key Formula or Approach:
There is a standard integral formula:
\[ \int e^{ax} [a \cdot f(x) + f'(x)] dx = e^{ax} f(x) + C \] This formula is derived from the product rule for differentiation: \( \frac{d}{dx}(e^{ax}f(x)) = a \cdot e^{ax}f(x) + e^{ax}f'(x) = e^{ax}(af(x) + f'(x)) \). We need to see if our integrand fits this pattern.
Step 3: Detailed Explanation:
The given integral is \( \int e^{2\theta} (2\cos^2\theta - \sin 2\theta) d\theta \).
By comparing with the formula \( \int e^{a\theta} [a \cdot f(\theta) + f'(\theta)] d\theta \), we can identify \( a=2 \).
So we are looking for a function \( f(\theta) \) such that the expression in the parenthesis is equal to \( 2f(\theta) + f'(\theta) \).
Let's test the functions that appear in the options. A good candidate for \( f(\theta) \) is \( \cos^2\theta \), as suggested by option A.
Let \( f(\theta) = \cos^2\theta \).
Now, let's find its derivative, \( f'(\theta) \).
Using the chain rule:
\[ f'(\theta) = \frac{d}{d\theta}(\cos\theta)^2 = 2(\cos\theta)^1 \cdot (-\sin\theta) = -2\sin\theta\cos\theta \] Using the double angle identity, \( \sin 2\theta = 2\sin\theta\cos\theta \), we have:
\[ f'(\theta) = -\sin 2\theta \] Now let's construct the expression \( a f(\theta) + f'(\theta) \) with \( a=2 \) and our chosen \( f(\theta) \):
\[ 2 f(\theta) + f'(\theta) = 2(\cos^2\theta) + (-\sin 2\theta) = 2\cos^2\theta - \sin 2\theta \] This perfectly matches the trigonometric part of our integrand.
Therefore, the integral fits the standard form.
The result of the integration is \( e^{a\theta} f(\theta) + C \).
Substituting our values:
\[ e^{2\theta} \cos^2\theta + C \] Step 4: Final Answer:
The integral is \( e^{2\theta} \cos^2\theta + C \).