`int x^2 sin^-1 x dx`

Answer

To solve the integral \( \int x^2 \sin^{-1}(x) \, dx \), we will use the method of integration by parts. The formula for integration by parts is given by:

\[ \int u \, dv = uv - \int v \, du \]

### Step 1: Choose \( u \) and \( dv \)

Let:
- \( u = \sin^{-1}(x) \) (which we will differentiate)
- \( dv = x^2 \, dx \) (which we will integrate)

### Step 2: Differentiate \( u \) and Integrate \( dv \)

Now we need to find \( du \) and \( v \):

1. Differentiate \( u \):
\[ du = \frac{1}{\sqrt{1 - x^2}} \, dx \]

2. Integrate \( dv \):
\[ v = \int x^2 \, dx = \frac{x^3}{3} \]

### Step 3: Apply the Integration by Parts Formula

Now we can apply the integration by parts formula:
\[ \int x^2 \sin^{-1}(x) \, dx = uv - \int v \, du \]

Substituting the values we found:
\[ = \sin^{-1}(x) \cdot \frac{x^3}{3} - \int \frac{x^3}{3} \cdot \frac{1}{\sqrt{1 - x^2}} \, dx \]

### Step 4: Simplify the Integral

This gives us:
\[ = \frac{x^3}{3} \sin^{-1}(x) - \frac{1}{3} \int \frac{x^3}{\sqrt{1 - x^2}} \, dx \]

### Step 5: Solve the Remaining Integral

To solve \( \int \frac{x^3}{\sqrt{1 - x^2}} \, dx \), we can use the substitution:
Let \( t = 1 - x^2 \), then \( dt = -2x \, dx \) or \( dx = -\frac{dt}{2x} \).

From \( t = 1 - x^2 \), we have \( x^2 = 1 - t \) and \( x^3 = x \cdot x^2 = x(1 - t) \).

Substituting \( x = \sqrt{1 - t} \):
\[ \int \frac{x^3}{\sqrt{1 - x^2}} \, dx = \int \frac{\sqrt{1 - t}(1 - t)}{\sqrt{t}} \cdot \left(-\frac{dt}{2\sqrt{1 - t}}\right) \]
This simplifies to:
\[ -\frac{1}{2} \int \frac{(1 - t)}{\sqrt{t}} \, dt \]

### Step 6: Integrate

Now, we can split the integral:
\[ -\frac{1}{2} \left( \int t^{-1/2} \, dt - \int t^{1/2} \, dt \right) \]
Calculating these integrals:
\[ -\frac{1}{2} \left( 2\sqrt{t} - \frac{2}{3} t^{3/2} \right) = -\sqrt{t} + \frac{1}{3} t^{3/2} \]

### Step 7: Substitute Back

Substituting back \( t = 1 - x^2 \):
\[ -\sqrt{1 - x^2} + \frac{1}{3} (1 - x^2)^{3/2} \]

### Final Result

Putting everything together:
\[ \int x^2 \sin^{-1}(x) \, dx = \frac{x^3}{3} \sin^{-1}(x) + \frac{1}{6} \left( -\sqrt{1 - x^2} + \frac{1}{3} (1 - x^2)^{3/2} \right) + C \]