Let `g(x)` be differentiable on `R` and `int_(sint)^1x^2g(x)dx=(1-sint),` where `t in (0,pi/2)dot` Then the value of `g(1/(sqrt(2)))` is____

Explanation

To find the value of \( g\left(\frac{1}{\sqrt{2}}\right) \) given the equation

\[ \int_{\sin t}^{1} x^2 g(x) \, dx = 1 - \sin t, \]

where \( t \in (0, \frac{\pi}{2}) \), we can follow these steps:

### Step 1: Differentiate both sides with respect to \( t \)

We apply the Fundamental Theorem of Calculus and differentiate both sides of the equation with respect to \( t \):

\[ \frac{d}{dt} \left( \int_{\sin t}^{1} x^2 g(x) \, dx \right) = \frac{d}{dt} (1 - \sin t). \]

### Step 2: Apply Leibniz's Rule

Using Leibniz's rule for differentiation under the integral sign, we have:

\[ \frac{d}{dt} \left( \int_{\sin t}^{1} x^2 g(x) \, dx \right) = x^2 g(x) \bigg|_{x=1} \cdot 0 - x^2 g(x) \bigg|_{x=\sin t} \cdot \cos t. \]

Since the upper limit is a constant (1), its derivative is 0. Thus, we only need to consider the lower limit:

\[ = -\sin^2 t \cdot g(\sin t) \cdot \cos t. \]

### Step 3: Differentiate the right side

Now differentiate the right side:

\[ \frac{d}{dt} (1 - \sin t) = -\cos t. \]

### Step 4: Set the derivatives equal

Setting the derivatives equal gives us:

\[ -\sin^2 t \cdot g(\sin t) \cdot \cos t = -\cos t. \]

### Step 5: Simplify the equation

We can cancel \(-\cos t\) from both sides (since \( \cos t \neq 0 \) for \( t \in (0, \frac{\pi}{2}) \)):

\[ \sin^2 t \cdot g(\sin t) = 1. \]

### Step 6: Solve for \( g(\sin t) \)

Thus, we have:

\[ g(\sin t) = \frac{1}{\sin^2 t}. \]

### Step 7: Find \( g\left(\frac{1}{\sqrt{2}}\right) \)

To find \( g\left(\frac{1}{\sqrt{2}}\right) \), we need to determine \( t \) such that \( \sin t = \frac{1}{\sqrt{2}} \). This occurs when \( t = \frac{\pi}{4} \).

Substituting \( t = \frac{\pi}{4} \) into our expression for \( g(\sin t) \):

\[ g\left(\frac{1}{\sqrt{2}}\right) = g(\sin(\frac{\pi}{4})) = \frac{1}{\sin^2(\frac{\pi}{4})}. \]

Since \( \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \):

\[ g\left(\frac{1}{\sqrt{2}}\right) = \frac{1}{\left(\frac{1}{\sqrt{2}}\right)^2} = \frac{1}{\frac{1}{2}} = 2. \]

### Final Answer

Thus, the value of \( g\left(\frac{1}{\sqrt{2}}\right) \) is

\[ \boxed{2}. \]