To solve the problem step by step, we need to find the value of the integral \( \int_{1}^{4} f'(x) \, dx \) given the conditions on the function \( f \).
### Step 1: Define the integral
Let:
\[
I = \int_{1}^{4} f'(x) \, dx
\]
### Step 2: Use integration by parts
We can use integration by parts, where we let \( u = x \) and \( dv = f''(x) \, dx \). Then, we have:
\[
du = dx \quad \text{and} \quad v = f'(x)
\]
Thus, by integration by parts:
\[
I = \left[ x f'(x) \right]_{1}^{4} - \int_{1}^{4} x f''(x) \, dx
\]
### Step 3: Evaluate the boundary term
Now we need to evaluate the boundary term:
\[
\left[ x f'(x) \right]_{1}^{4} = 4f'(4) - 1f'(1)
\]
### Step 4: Define a new integral
Let:
\[
J = \int_{1}^{4} x f''(x) \, dx
\]
We can express \( J \) using the property of \( f''(x) \):
\[
J = \int_{1}^{4} x f''(x) \, dx = \int_{1}^{4} (5 - x) f''(5 - x) \, dx
\]
### Step 5: Change of variables
Using the substitution \( u = 5 - x \), we have \( du = -dx \). The limits change as follows:
- When \( x = 1 \), \( u = 4 \)
- When \( x = 4 \), \( u = 1 \)
Thus, we can rewrite \( J \):
\[
J = \int_{4}^{1} (5 - (5 - u)) f''(u) (-du) = \int_{1}^{4} (5 - u) f''(u) \, du
\]
This gives us:
\[
J = \int_{1}^{4} (5 - u) f''(u) \, du
\]
### Step 6: Combine the integrals
Now we have:
\[
J = \int_{1}^{4} 5 f''(u) \, du - \int_{1}^{4} u f''(u) \, du
\]
Thus:
\[
2J = 5 \int_{1}^{4} f''(u) \, du - J
\]
This implies:
\[
3J = 5 \int_{1}^{4} f''(u) \, du
\]
So:
\[
J = \frac{5}{3} \int_{1}^{4} f''(u) \, du
\]
### Step 7: Evaluate the integral of \( f''(x) \)
Using the Fundamental Theorem of Calculus:
\[
\int_{1}^{4} f''(x) \, dx = f'(4) - f'(1)
\]
### Step 8: Substitute back into the equation for \( I \)
Now substituting back:
\[
I = 4f'(4) - f'(1) - J
\]
Substituting \( J \):
\[
I = 4f'(4) - f'(1) - \frac{5}{3}(f'(4) - f'(1))
\]
### Step 9: Solve for \( I \)
Now we can combine terms:
\[
I = 4f'(4) - f'(1) - \frac{5}{3}f'(4) + \frac{5}{3}f'(1)
\]
This simplifies to:
\[
I = \left(4 - \frac{5}{3}\right)f'(4) + \left(-1 + \frac{5}{3}\right)f'(1)
\]
Calculating the coefficients:
\[
I = \frac{12 - 5}{3}f'(4) + \frac{-3 + 5}{3}f'(1) = \frac{7}{3}f'(4) + \frac{2}{3}f'(1)
\]
### Step 10: Substitute values for \( f'(0) \) and \( f'(5) \)
From the problem, we know \( f'(0) = 1 \) and \( f'(5) = 7 \). By symmetry and the property of \( f''(x) \):
\[
f'(4) + f'(1) = 8
\]
Let \( f'(4) = a \) and \( f'(1) = b \). Then:
\[
a + b = 8
\]
Substituting \( b = 8 - a \) into \( I \):
\[
I = \frac{7}{3}a + \frac{2}{3}(8 - a) = \frac{7a + 16 - 2a}{3} = \frac{5a + 16}{3}
\]
### Step 11: Solve for \( a \) and \( b \)
Using \( f'(4) + f'(1) = 8 \) and the values of \( f'(0) \) and \( f'(5) \):
Assuming \( f'(4) = 7 \) and \( f'(1) = 1 \) gives:
\[
I = \frac{5(7) + 16}{3} = \frac{35 + 16}{3} = \frac{51}{3} = 17
\]
### Conclusion
After evaluating and simplifying, we find:
\[
\int_{1}^{4} f'(x) \, dx = 8
\]
### Final Answer
Thus, the value of \( \int_{1}^{4} f'(x) \, dx \) is:
\[
\boxed{8}
\]
