If `f(0)=0, f(3)=3 and f'(3)=4`, then the value of `int_(0)^(1)xf'' (3x)dx` is equal to

Answer

To solve the problem, we need to evaluate the integral \( I = \int_{0}^{1} x f''(3x) \, dx \) given the conditions \( f(0) = 0 \), \( f(3) = 3 \), and \( f'(3) = 4 \).

### Step-by-Step Solution:

1. **Substitution**:
Let's perform a substitution. Let \( u = 3x \). Then, \( du = 3 \, dx \) or \( dx = \frac{du}{3} \). The limits change as follows:
- When \( x = 0 \), \( u = 0 \).
- When \( x = 1 \), \( u = 3 \).

Thus, the integral becomes:
\[ I = \int_{0}^{3} \frac{u}{3} f''(u) \cdot \frac{du}{3} = \frac{1}{9} \int_{0}^{3} u f''(u) \, du. \]

**Hint**: Use substitution to simplify the integral.

2. **Integration by Parts**:
We will use integration by parts on the integral \( \int u f''(u) \, du \). Let:
- \( v = f'(u) \) (thus \( dv = f''(u) \, du \))
- \( w = u \) (thus \( dw = du \))

By the integration by parts formula \( \int v \, dw = vw - \int w \, dv \), we have:
\[ \int u f''(u) \, du = u f'(u) \bigg|_{0}^{3} - \int f'(u) \, du. \]

**Hint**: Remember the integration by parts formula and apply it correctly.

3. **Evaluate the Boundary Terms**:
Now we evaluate \( u f'(u) \) at the limits:
\[ u f'(u) \bigg|_{0}^{3} = 3 f'(3) - 0 \cdot f'(0) = 3 \cdot 4 = 12. \]

**Hint**: Substitute the values given in the problem for \( f'(3) \).

4. **Evaluate the Remaining Integral**:
Now we need to evaluate \( \int f'(u) \, du \):
\[ \int f'(u) \, du = f(u) \bigg|_{0}^{3} = f(3) - f(0) = 3 - 0 = 3. \]

**Hint**: Use the Fundamental Theorem of Calculus to evaluate the integral of \( f' \).

5. **Combine the Results**:
Now we can combine the results:
\[ \int u f''(u) \, du = 12 - 3 = 9. \]

6. **Final Calculation**:
Recall that we had \( I = \frac{1}{9} \int_{0}^{3} u f''(u) \, du \):
\[ I = \frac{1}{9} \cdot 9 = 1. \]

### Conclusion:
Thus, the value of the integral \( \int_{0}^{1} x f''(3x) \, dx \) is \( \boxed{1} \).