Question

Evaluate: \[ \int \frac{2 + \sin 2x}{1 + \cos 2x} e^x \, dx. \]

Hint:

For trigonometric integrals, simplify using identities before integrating.

Answer 1

Step 1: Simplify the integrand
Use the identity \( 1 + \cos 2x = 2\cos^2 x \) and \( \sin 2x = 2\sin x \cos x \): \[ \frac{2 + \sin 2x}{1 + \cos 2x} = \frac{2 + 2\sin x \cos x}{2\cos^2 x} = \sec^2 x + \tan x. \] Step 2: Rewrite the integral
\[ I = \int (\sec^2 x + \tan x) e^x \, dx. \] Step 3: Integrate term by term
For \( \int \sec^2 x e^x \, dx \), use substitution \( u = \tan x \): \[ \int \sec^2 x e^x \, dx = e^x \tan x. \] For \( \int \tan x e^x \, dx \), combine it with the first term: \[ I = e^x \tan x + C. \]