To solve the integral \( \int e^{\sin \theta} (\sin \theta + \sec^2 \theta) \, d\theta \) and find the value of \( f\left(\frac{\pi}{4}\right) \) given that \( f(0) = 0 \), we can follow these steps:
### Step 1: Split the Integral
We can split the integral into two parts:
\[
\int e^{\sin \theta} (\sin \theta + \sec^2 \theta) \, d\theta = \int e^{\sin \theta} \sin \theta \, d\theta + \int e^{\sin \theta} \sec^2 \theta \, d\theta
\]
### Step 2: Solve the Second Integral Using Integration by Parts
For the second integral \( \int e^{\sin \theta} \sec^2 \theta \, d\theta \), we can use integration by parts. Let:
- \( u = e^{\sin \theta} \) (first function)
- \( dv = \sec^2 \theta \, d\theta \) (second function)
Then, we find:
- \( du = e^{\sin \theta} \cos \theta \, d\theta \)
- \( v = \tan \theta \)
Using integration by parts:
\[
\int u \, dv = uv - \int v \, du
\]
we have:
\[
\int e^{\sin \theta} \sec^2 \theta \, d\theta = e^{\sin \theta} \tan \theta - \int \tan \theta (e^{\sin \theta} \cos \theta) \, d\theta
\]
### Step 3: Simplify the Expression
The integral \( \int \tan \theta (e^{\sin \theta} \cos \theta) \, d\theta \) can be rewritten as:
\[
\int e^{\sin \theta} \sin \theta \, d\theta
\]
Thus, we have:
\[
\int e^{\sin \theta} \sec^2 \theta \, d\theta = e^{\sin \theta} \tan \theta - \int e^{\sin \theta} \sin \theta \, d\theta
\]
### Step 4: Combine the Results
Now, we can combine the results:
\[
\int e^{\sin \theta} (\sin \theta + \sec^2 \theta) \, d\theta = \int e^{\sin \theta} \sin \theta \, d\theta + \left( e^{\sin \theta} \tan \theta - \int e^{\sin \theta} \sin \theta \, d\theta \right)
\]
This simplifies to:
\[
\int e^{\sin \theta} (\sin \theta + \sec^2 \theta) \, d\theta = e^{\sin \theta} \tan \theta + C
\]
### Step 5: Identify \( f(\theta) \)
From the integral, we can identify:
\[
f(\theta) = e^{\sin \theta} \tan \theta
\]
### Step 6: Evaluate \( f\left(\frac{\pi}{4}\right) \)
Now we need to find \( f\left(\frac{\pi}{4}\right) \):
\[
f\left(\frac{\pi}{4}\right) = e^{\sin\left(\frac{\pi}{4}\right)} \tan\left(\frac{\pi}{4}\right)
\]
Calculating the values:
- \( \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \)
- \( \tan\left(\frac{\pi}{4}\right) = 1 \)
Thus:
\[
f\left(\frac{\pi}{4}\right) = e^{\frac{1}{\sqrt{2}}} \cdot 1 = e^{\frac{1}{\sqrt{2}}}
\]
### Final Answer
The value of \( f\left(\frac{\pi}{4}\right) \) is:
\[
\boxed{e^{\frac{1}{\sqrt{2}}}}
\]
