If the middle term of ` (x + 1/x"sin"^(-1)x)^(8) " is equal to (630)/(16)` ,
the value of x is/are

Options

A. `- (pi)/(3)`
B. `-(pi)/(6)`
C. `(pi)/(6)`
D. `(pi)/(3)`

Explanation

To solve the problem, we need to find the value of \( x \) such that the middle term of the expansion \( (x + \frac{1}{x} \sin^{-1} x)^8 \) is equal to \( \frac{630}{16} \).

### Step 1: Identify the middle term in the binomial expansion
For the binomial expansion of \( (a + b)^n \), the middle term is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
where \( n \) is the power of the binomial, and \( r = \frac{n}{2} \) if \( n \) is even.

Here, \( n = 8 \), so the middle term (5th term) is:
\[ T_5 = \binom{8}{4} \left(x\right^{8-4} \left(\frac{1}{x} \sin^{-1} x\right)^4 \]

### Step 2: Calculate \( T_5 \)
Substituting \( n = 8 \) and \( r = 4 \):
\[ T_5 = \binom{8}{4} x^4 \left(\frac{1}{x} \sin^{-1} x\right)^4 \]
This simplifies to:
\[ T_5 = \binom{8}{4} x^4 \cdot \frac{1}{x^4} \cdot (\sin^{-1} x)^4 = \binom{8}{4} (\sin^{-1} x)^4 \]

### Step 3: Calculate \( \binom{8}{4} \)
Calculating \( \binom{8}{4} \):
\[ \binom{8}{4} = \frac{8!}{4! \cdot 4!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]

### Step 4: Set up the equation
Now we have:
\[ T_5 = 70 (\sin^{-1} x)^4 \]
We set this equal to \( \frac{630}{16} \):
\[ 70 (\sin^{-1} x)^4 = \frac{630}{16} \]

### Step 5: Simplify the equation
Dividing both sides by 70:
\[ (\sin^{-1} x)^4 = \frac{630}{16 \times 70} \]
Calculating the right side:
\[ \frac{630}{1120} = \frac{9}{16} \]
Thus, we have:
\[ (\sin^{-1} x)^4 = \frac{9}{16} \]

### Step 6: Solve for \( \sin^{-1} x \)
Taking the fourth root:
\[ \sin^{-1} x = \pm \frac{3}{2} \]

### Step 7: Find \( x \)
Since \( \sin^{-1} x \) must be within the range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), we find:
\[ \sin^{-1} x = \frac{\pi}{3} \quad \text{or} \quad \sin^{-1} x = -\frac{\pi}{3} \]
This gives us:
\[ x = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \quad \text{and} \quad x = \sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} \]

### Final Answer
Thus, the values of \( x \) are:
\[ x = \frac{\sqrt{3}}{2} \quad \text{and} \quad x = -\frac{\sqrt{3}}{2} \]