Question:
\( \int e^{x}\left( \frac{2+\sin 2x}{1+\cos 2x} \right) dx \) is equal to :
\( \int e^{x}\left( \frac{2+\sin 2x}{1+\cos 2x} \right) dx \) is equal to :
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Simplify the fraction to sec^2 x + tan x and use the rule for e^x times (f(x) plus the derivative of f(x)) with f(x) = tan x.
Updated On: Oct 22, 2026
- \( \log \cos 2x + C \)
- \( \sec^{2} 2x + C \)
- \( e^{x}\tan x + C \)
- \( e^{x}\cot x + C \)
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The Correct Option is C
Solution and Explanation
Step 1: Simplify the fraction:
Use \( 1 + \cos 2x = 2\cos^2 x \) and \( \sin 2x = 2\sin x \cos x \).
\[ \frac{2+\sin 2x}{1+\cos 2x} = \frac{2 + 2\sin x\cos x}{2\cos^2 x} = \frac{1}{\cos^2 x} + \frac{\sin x}{\cos x} = \sec^2 x + \tan x \]
Step 2: Match the standard form of e^x times (f + derivative of f):
Let \( f(x) = \tan x \). Then \( f^{\prime}(x) = \sec^2 x \).
So the integrand is \( e^{x}\left[ f(x) + f^{\prime}(x) \right] \).
Step 3: Use the standard result:
We know \( \int e^{x}\left[ f(x) + f^{\prime}(x) \right] dx = e^{x} f(x) + C \).
\[ \int e^{x}(\tan x + \sec^2 x)\, dx = e^{x}\tan x + C \]
Step 4: Why the other options fail:
Option (A) has no factor \( e^x \), so its derivative cannot contain \( e^x \).
Option (B) also has no \( e^x \) factor.
Option (D): the derivative of \( e^x \cot x \) is \( e^x(\cot x - \operatorname{cosec}^2 x) \), which is not the integrand.
Final Answer:
The integral is \( e^x \tan x + C \), option (C).
\[ \boxed{e^{x}\tan x + C} \]
Use \( 1 + \cos 2x = 2\cos^2 x \) and \( \sin 2x = 2\sin x \cos x \).
\[ \frac{2+\sin 2x}{1+\cos 2x} = \frac{2 + 2\sin x\cos x}{2\cos^2 x} = \frac{1}{\cos^2 x} + \frac{\sin x}{\cos x} = \sec^2 x + \tan x \]
Step 2: Match the standard form of e^x times (f + derivative of f):
Let \( f(x) = \tan x \). Then \( f^{\prime}(x) = \sec^2 x \).
So the integrand is \( e^{x}\left[ f(x) + f^{\prime}(x) \right] \).
Step 3: Use the standard result:
We know \( \int e^{x}\left[ f(x) + f^{\prime}(x) \right] dx = e^{x} f(x) + C \).
\[ \int e^{x}(\tan x + \sec^2 x)\, dx = e^{x}\tan x + C \]
Step 4: Why the other options fail:
Option (A) has no factor \( e^x \), so its derivative cannot contain \( e^x \).
Option (B) also has no \( e^x \) factor.
Option (D): the derivative of \( e^x \cot x \) is \( e^x(\cot x - \operatorname{cosec}^2 x) \), which is not the integrand.
Final Answer:
The integral is \( e^x \tan x + C \), option (C).
\[ \boxed{e^{x}\tan x + C} \]
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