`int_(0)^(pi//4) e^(x) (tan x+ sec^(2)x) dx`

Answer

To solve the integral \(\int_{0}^{\frac{\pi}{4}} e^{x} \left( \tan x + \sec^{2} x \right) dx\), we can use the integration technique that involves recognizing the derivative of a function within the integrand.

### Step-by-Step Solution:

1. **Identify the Function and its Derivative**:
We notice that the integrand consists of \(e^{x}\) multiplied by \(\tan x + \sec^{2} x\). We know that the derivative of \(\tan x\) is \(\sec^{2} x\). Therefore, we can express the integrand as:
\[ \tan x + \sec^{2} x = \frac{d}{dx}(\tan x) + \tan x \]

2. **Recognize the Integration Formula**:
There is a useful integration formula:
\[ \int e^{x} (f(x) + f'(x)) \, dx = e^{x} f(x) + C \]
Here, \(f(x) = \tan x\) and \(f'(x) = \sec^{2} x\).

3. **Apply the Formula**:
We can rewrite the integral:
\[ \int_{0}^{\frac{\pi}{4}} e^{x} \left( \tan x + \sec^{2} x \right) dx = \int_{0}^{\frac{\pi}{4}} e^{x} \frac{d}{dx}(\tan x) + e^{x} \tan x \, dx \]
This simplifies to:
\[ \int_{0}^{\frac{\pi}{4}} e^{x} \frac{d}{dx}(\tan x) \, dx = e^{x} \tan x \bigg|_{0}^{\frac{\pi}{4}} \]

4. **Evaluate the Limits**:
Now we need to evaluate the expression \(e^{x} \tan x\) at the limits \(0\) and \(\frac{\pi}{4}\):
\[ e^{\frac{\pi}{4}} \tan\left(\frac{\pi}{4}\right) - e^{0} \tan(0) \]
We know that \(\tan\left(\frac{\pi}{4}\right) = 1\) and \(\tan(0) = 0\):
\[ = e^{\frac{\pi}{4}} \cdot 1 - e^{0} \cdot 0 = e^{\frac{\pi}{4}} - 0 = e^{\frac{\pi}{4}} \]

5. **Final Answer**:
Therefore, the value of the integral is:
\[ \int_{0}^{\frac{\pi}{4}} e^{x} \left( \tan x + \sec^{2} x \right) dx = e^{\frac{\pi}{4}} \]