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To solve for \( f(x) \) given the conditions:

1. \( f''(x) = \sin x + \cos x \)
2. \( f'(0) = 5 \)
3. \( f(0) = 5 \)

we'll proceed by integrating step-by-step.

### Step 1: Find \( f'(x) \) by integrating \( f''(x) \).
Given:
\[
f''(x) = \sin x + \cos x
\]
Integrate \( f''(x) \) with respect to \( x \):
\[
f'(x) = \int (\sin x + \cos x) \, dx
\]
\[
f'(x) = -\cos x + \sin x + C_1
\]
where \( C_1 \) is the constant of integration.

### Step 2: Use \( f'(0) = 5 \) to solve for \( C_1 \).
Substitute \( x = 0 \) into \( f'(x) \):
\[
f'(0) = -\cos(0) + \sin(0) + C_1 = -1 + 0 + C_1 = 5
\]
\[
C_1 = 6
\]
So,
\[
f'(x) = -\cos x + \sin x + 6
\]

### Step 3: Find \( f(x) \) by integrating \( f'(x) \).
Now integrate \( f'(x) \):
\[
f(x) = \int (-\cos x + \sin x + 6) \, dx
\]
\[
f(x) = -\sin x - \cos x + 6x + C_2
\]
where \( C_2 \) is another constant of integration.

### Step 4: Use \( f(0) = 5 \) to solve for \( C_2 \).
Substitute \( x = 0 \) into \( f(x) \):
\[
f(0) = -\sin(0) - \cos(0) + 6 \cdot 0 + C_2 = 0 - 1 + 0 + C_2 = 5
\]
\[
C_2 = 6
\]
So,
\[
f(x) = -\sin x - \cos x + 6x + 6
\]

### Final Answer
\[
f(x) = -\sin x - \cos x + 6x + 6
\]

Would you like more details on any step, or have any further questions?

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### Additional Related Questions:
1. How would the answer change if \( f(0) \) were a different value?
2. What if \( f'(x) \) had additional terms beyond trigonometric functions?
3. Can we use alternative methods, like Laplace transforms, for such problems?
4. How can we verify our solution by differentiating \( f(x) \) back?
5. What if \( f(x) \) were required to satisfy an initial condition at a different point?

### Tip:
Remember to check initial conditions carefully, as they determine the constants in integration solutions.

Find f if f''(x) = sin x + cos x, f'(0) = 5, and f(0) = 5.

Learn how to find the function f(x) by integrating f''(x) = sin x + cos x with initial conditions f'(0) = 5 and f(0) = 5. This solution covers differentiation, integration, and applying initial conditions, suitable for high school students in Grades 11-12.

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