The value of `int(xsin^(-1)x)/(sqrt((1-x^2)))` dx is

Options

A. `sqrt((1-x^2))sin^(-1)x`
B. `xsin^(-1)x`
C. `x-sqrt((1-x^2))sin^(-1)x`
D. `(sin^(-1)x)^2`

Answer

To solve the integral \( \int \frac{x \sin^{-1}(x)}{\sqrt{1 - x^2}} \, dx \), we will use integration by parts. Let's break it down step by step.

### Step 1: Identify Parts for Integration by Parts
We will choose:
- \( u = \sin^{-1}(x) \) (which we will differentiate)
- \( dv = \frac{x}{\sqrt{1 - x^2}} \, dx \) (which we will integrate)

### Step 2: Differentiate and Integrate
Now, we need to find \( du \) and \( v \):
- Differentiate \( u \):
\[ du = \frac{1}{\sqrt{1 - x^2}} \, dx \]

- Integrate \( dv \):
To integrate \( dv = \frac{x}{\sqrt{1 - x^2}} \, dx \), we can use the substitution \( t = 1 - x^2 \), which gives \( dt = -2x \, dx \) or \( dx = -\frac{dt}{2x} \). Thus,
\[ v = \int \frac{x}{\sqrt{1 - x^2}} \, dx = -\frac{1}{2} \sqrt{1 - x^2} \]

### Step 3: Apply Integration by Parts Formula
Using the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
we substitute our values:
\[ \int \frac{x \sin^{-1}(x)}{\sqrt{1 - x^2}} \, dx = \sin^{-1}(x) \left(-\frac{1}{2} \sqrt{1 - x^2}\right) - \int \left(-\frac{1}{2} \sqrt{1 - x^2}\right) \left(\frac{1}{\sqrt{1 - x^2}} \, dx\right) \]

### Step 4: Simplify the Expression
This simplifies to:
\[ -\frac{1}{2} \sin^{-1}(x) \sqrt{1 - x^2} + \frac{1}{2} \int dx \]
The integral \( \int dx \) is simply \( x \).

### Step 5: Combine the Results
Putting it all together:
\[ \int \frac{x \sin^{-1}(x)}{\sqrt{1 - x^2}} \, dx = -\frac{1}{2} \sin^{-1}(x) \sqrt{1 - x^2} + \frac{1}{2} x + C \]

### Final Answer
Thus, the value of the integral is:
\[ -\frac{1}{2} \sin^{-1}(x) \sqrt{1 - x^2} + \frac{1}{2} x + C \]