Evaluate`int e^(x) ((1+sinx cos x)/(cos^(2)x))dx`

Answer

To evaluate the integral

\[ \int e^x \left( \frac{1 + \sin x \cos x}{\cos^2 x} \right) dx, \]

we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integral:

\[ \int e^x \left( \frac{1}{\cos^2 x} + \frac{\sin x \cos x}{\cos^2 x} \right) dx. \]

This simplifies to:

\[ \int e^x \left( \sec^2 x + \tan x \right) dx. \]

### Step 2: Break Down the Integral
Now we can break this integral into two parts:

\[ \int e^x \sec^2 x \, dx + \int e^x \tan x \, dx. \]

### Step 3: Use Integration by Parts
For the integral \(\int e^x \tan x \, dx\), we can use the integration by parts formula:

\[ \int u \, dv = uv - \int v \, du. \]

Let:
- \(u = \tan x\) \(\Rightarrow du = \sec^2 x \, dx\)
- \(dv = e^x \, dx\) \(\Rightarrow v = e^x\)

Applying the integration by parts:

\[ \int e^x \tan x \, dx = e^x \tan x - \int e^x \sec^2 x \, dx. \]

### Step 4: Combine the Integrals
Now we can substitute back into our original integral:

\[ \int e^x \sec^2 x \, dx + \left( e^x \tan x - \int e^x \sec^2 x \, dx \right). \]

This simplifies to:

\[ e^x \tan x + \int e^x \sec^2 x \, dx - \int e^x \sec^2 x \, dx = e^x \tan x. \]

### Step 5: Final Answer
Thus, the final answer for the integral is:

\[ \int e^x \left( \frac{1 + \sin x \cos x}{\cos^2 x} \right) dx = e^x \tan x + C, \]

where \(C\) is the constant of integration.

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