Question

The integral $\int e^x \frac{2 + \sin 2x}{1 + \cos 2x} \, dx$ is equal to

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

Hint:

When dealing with integrals involving trigonometric functions, it’s useful to apply identities to simplify the integrand before attempting the integration.

Answer 1

The Correct Option is B

We are given the integral: $$I = \int e^x \frac{2 + \sin 2x}{1 + \cos 2x} \, dx.$$ Step 1: Simplify the expression inside the integral. Recall the trigonometric identity: $$1 + \cos 2x = 2\cos^2 x.$$ Substitute this into the denominator: $$I = \int e^x \frac{2 + \sin 2x}{2\cos^2 x} \, dx.$$ Next, separate the terms: $$I = \int e^x \frac{2}{2\cos^2 x} \, dx + \int e^x \frac{\sin 2x}{2\cos^2 x} \, dx.$$ Simplify the fractions: $$I = \int e^x \sec^2 x \, dx + \int e^x \tan x \sec x \, dx.$$ Step 2: Evaluate each integral. For the first term: $$\int e^x \sec^2 x \, dx,$$ we use the standard result: $$\int e^x \sec^2 x \, dx = e^x \tan x + C_1.$$ For the second term: $$\int e^x \tan x \sec x \, dx,$$ we apply the known formula: $$\int e^x \tan x \sec x \, dx = e^x \tan x + C_2.$$ Step 3: Combine the results. Adding both terms together: $$I = e^x \tan x + C.$$ Final Answer: $$\boxed{e^x \tan x + C}$$