The value of `int_(a)^(b)(x-a)^(3)(b-x)^(4)dx` is `((b-a)^(m))/(n).` Then `(m, n)` is

Options

A. (6, 260)
B. (8, 280)
C. (2, 240)
D. none of these

Answer

To solve the integral \( I = \int_{a}^{b} (x-a)^{3} (b-x)^{4} \, dx \), we can use the formula for the integral of the form \( \int_{a}^{b} (x-a)^{m} (b-x)^{n} \, dx \), which is given by:

\[ I_{m,n} = \frac{n! \, m!}{(m+n+1)!} (b-a)^{m+n+1} \]

In our case, we have \( m = 3 \) and \( n = 4 \).

### Step 1: Identify \( m \) and \( n \)
From the problem, we have:
- \( m = 3 \)
- \( n = 4 \)

### Step 2: Calculate \( m+n \)
Now, we calculate \( m+n \):
\[ m+n = 3 + 4 = 7 \]

### Step 3: Calculate \( m+n+1 \)
Next, we calculate \( m+n+1 \):
\[ m+n+1 = 7 + 1 = 8 \]

### Step 4: Calculate the factorials
Now we need to calculate the factorials:
- \( n! = 4! = 24 \)
- \( m! = 3! = 6 \)
- \( (m+n+1)! = 8! = 40320 \)

### Step 5: Substitute into the formula
Now we can substitute these values into the formula:
\[ I = \frac{4! \, 3!}{(3+4+1)!} (b-a)^{3+4+1} = \frac{24 \cdot 6}{40320} (b-a)^{8} \]

### Step 6: Simplify the fraction
Calculating the fraction:
\[ \frac{24 \cdot 6}{40320} = \frac{144}{40320} \]
Now, simplifying \( \frac{144}{40320} \):
\[ \frac{144}{40320} = \frac{1}{280} \]

### Step 7: Final expression for \( I \)
Thus, we have:
\[ I = \frac{(b-a)^{8}}{280} \]

### Step 8: Identify \( m \) and \( n \)
From the expression \( I = \frac{(b-a)^{m}}{n} \), we can identify:
- \( m = 8 \)
- \( n = 280 \)

### Conclusion
Thus, the values of \( (m, n) \) are \( (8, 280) \).