`int_0^x{int_0^uf(t)dx}\ du` is equal to (a) `int_0^x(x-u)f(u)du` (b) `int_0^x uf(x-u)du` (c) `x int_0^xf(u)du` (d) `x int_0^x uf(u-x)du`

Options

A. `int_(0)^(x)(x-u)f(u)du`
B. `int_(0)^(x) uf(x-u)du`
C. `x int_(0)^(x)f(u)du`
D. `x int_(0)^(x)uf(u-x)du`

Explanation

To solve the integral \( \int_0^x \left( \int_0^u f(t) \, dt \right) du \), we can follow these steps:

### Step 1: Understand the Structure of the Integral
We have a double integral where the outer integral is with respect to \( u \) and the inner integral is with respect to \( t \). The limits of the inner integral depend on \( u \).

### Step 2: Change the Order of Integration
We can change the order of integration. The region of integration is defined by \( 0 \leq t \leq u \) and \( 0 \leq u \leq x \). This means we can also express the integral as:
\[ \int_0^x \left( \int_t^x du \right) f(t) \, dt \]
Here, for a fixed \( t \), \( u \) ranges from \( t \) to \( x \).

### Step 3: Evaluate the Inner Integral
The inner integral \( \int_t^x du \) can be evaluated easily:
\[ \int_t^x du = x - t \]
Thus, we can rewrite the double integral as:
\[ \int_0^x (x - t) f(t) \, dt \]

### Step 4: Rewrite the Integral
Now we have:
\[ \int_0^x (x - t) f(t) \, dt \]
This expression can be rewritten in terms of \( u \) by substituting \( t \) with \( u \):
\[ \int_0^x (x - u) f(u) \, du \]

### Conclusion
Thus, the original integral simplifies to:
\[ \int_0^x (x - u) f(u) \, du \]
This matches option (a).

### Final Answer
The answer is:
\[ \int_0^x (x - u) f(u) \, du \quad \text{(Option a)} \]

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