Evaluate : `inte^(x).((2+sin2x))/(cos^(2)x)dx`

Answer

To evaluate the integral \( I = \int e^x \frac{2 + \sin 2x}{\cos^2 x} \, dx \), we will follow these steps:

### Step 1: Rewrite the integral
We start by rewriting the integral for clarity:
\[ I = \int e^x \frac{2 + \sin 2x}{\cos^2 x} \, dx \]

### Step 2: Simplify \(\sin 2x\)
Recall the double angle identity for sine:
\[ \sin 2x = 2 \sin x \cos x \]
Substituting this into the integral gives:
\[ I = \int e^x \frac{2 + 2 \sin x \cos x}{\cos^2 x} \, dx \]

### Step 3: Factor out the constant
We can factor out the 2 from the numerator:
\[ I = \int e^x \left( \frac{2}{\cos^2 x} + \frac{2 \sin x \cos x}{\cos^2 x} \right) \, dx \]
This simplifies to:
\[ I = 2 \int e^x \left( \sec^2 x + \tan x \right) \, dx \]

### Step 4: Split the integral
Now we can split the integral into two parts:
\[ I = 2 \left( \int e^x \sec^2 x \, dx + \int e^x \tan x \, dx \right) \]

### Step 5: Use integration by parts
For the integral \( \int e^x \tan x \, dx \), we can use integration by parts. Let:
- \( u = \tan x \) ⇒ \( du = \sec^2 x \, dx \)
- \( dv = e^x \, dx \) ⇒ \( v = e^x \)

Using integration by parts:
\[ \int e^x \tan x \, dx = e^x \tan x - \int e^x \sec^2 x \, dx \]

### Step 6: Substitute back into the integral
Now substituting back into our expression for \( I \):
\[ I = 2 \left( \int e^x \sec^2 x \, dx + e^x \tan x - \int e^x \sec^2 x \, dx \right) \]
The \( \int e^x \sec^2 x \, dx \) terms cancel out:
\[ I = 2 e^x \tan x + C \]

### Final Result
Thus, the evaluated integral is:
\[ I = 2 e^x \tan x + C \]