`int_(0)^(1//sqrt(2))(x sin^(-1)x)/(sqrt(1-x^(2)))dx=`

Options

A. `(pi)/(4)+(2)/(sqrt(2))`
B. `(pi+4)/(4sqrt(2))`
C. `(pi)/(4)-(2)/(sqrt(2))`
D. `(pi)/(4)-(1)/(sqrt(2))`

Answer

To solve the integral

\[ I = \int_{0}^{\frac{1}{\sqrt{2}}} \frac{x \sin^{-1} x}{\sqrt{1 - x^2}} \, dx, \]

we will use the substitution method and integration by parts.

### Step 1: Substitution
Let \( t = \sin^{-1} x \). Then, we have:

\[ x = \sin t \quad \text{and} \quad dx = \cos t \, dt. \]

Also, we need to change the limits of integration:
- When \( x = 0 \), \( t = \sin^{-1}(0) = 0 \).
- When \( x = \frac{1}{\sqrt{2}} \), \( t = \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4} \).

Thus, the integral becomes:

\[ I = \int_{0}^{\frac{\pi}{4}} \frac{\sin t \cdot t}{\sqrt{1 - \sin^2 t}} \cos t \, dt. \]

Since \( \sqrt{1 - \sin^2 t} = \cos t \), we can simplify the integral:

\[ I = \int_{0}^{\frac{\pi}{4}} t \sin t \, dt. \]

### Step 2: Integration by Parts
Now we will use integration by parts. Let:

- \( u = t \) (then \( du = dt \))
- \( dv = \sin t \, dt \) (then \( v = -\cos t \))

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \):

\[ I = \left[ -t \cos t \right]_{0}^{\frac{\pi}{4}} - \int_{0}^{\frac{\pi}{4}} -\cos t \, dt. \]

### Step 3: Evaluate the Boundary Terms
Now we evaluate the boundary terms:

\[ \left[ -t \cos t \right]_{0}^{\frac{\pi}{4}} = -\left(\frac{\pi}{4} \cdot \cos\left(\frac{\pi}{4}\right) - 0 \cdot \cos(0)\right) = -\left(\frac{\pi}{4} \cdot \frac{1}{\sqrt{2}}\right) = -\frac{\pi}{4\sqrt{2}}. \]

### Step 4: Evaluate the Remaining Integral
Now we compute the remaining integral:

\[ \int_{0}^{\frac{\pi}{4}} \cos t \, dt = \left[ \sin t \right]_{0}^{\frac{\pi}{4}} = \sin\left(\frac{\pi}{4}\right) - \sin(0) = \frac{1}{\sqrt{2}} - 0 = \frac{1}{\sqrt{2}}. \]

### Step 5: Combine the Results
Putting it all together:

\[ I = -\frac{\pi}{4\sqrt{2}} + \frac{1}{\sqrt{2}}. \]

To combine these, we find a common denominator:

\[ I = \frac{-\pi + 4}{4\sqrt{2}}. \]

Thus, the final result is:

\[ I = \frac{4 - \pi}{4\sqrt{2}}. \]

### Final Answer
\[ \int_{0}^{\frac{1}{\sqrt{2}}} \frac{x \sin^{-1} x}{\sqrt{1 - x^2}} \, dx = \frac{4 - \pi}{4\sqrt{2}}. \]