Evaluate the following integrals:
`intsin^(-1)sqrt(x)dx`

Explanation

To evaluate the integral \( \int \sin^{-1}(\sqrt{x}) \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( \sqrt{x} = \sin(t) \). Then, we have:
\[ x = \sin^2(t) \quad \text{and} \quad dx = 2\sin(t)\cos(t) \, dt = \sin(2t) \, dt \]

### Step 2: Change of Variables
Substituting \( \sqrt{x} \) and \( dx \) into the integral, we get:
\[ \int \sin^{-1}(\sqrt{x}) \, dx = \int \sin^{-1}(\sin(t)) \cdot \sin(2t) \, dt \]
Since \( \sin^{-1}(\sin(t)) = t \) for \( t \) in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \), we can simplify the integral to:
\[ \int t \sin(2t) \, dt \]

### Step 3: Integration by Parts
Now we will use integration by parts. Let:
- \( u = t \) (thus \( du = dt \))
- \( dv = \sin(2t) \, dt \) (thus \( v = -\frac{1}{2}\cos(2t) \))

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \), we have:
\[ \int t \sin(2t) \, dt = -\frac{1}{2} t \cos(2t) - \int -\frac{1}{2} \cos(2t) \, dt \]

### Step 4: Evaluate the Remaining Integral
Now we need to evaluate \( \int \cos(2t) \, dt \):
\[ \int \cos(2t) \, dt = \frac{1}{2} \sin(2t) \]
Thus, substituting back, we get:
\[ \int t \sin(2t) \, dt = -\frac{1}{2} t \cos(2t) + \frac{1}{4} \sin(2t) + C \]

### Step 5: Back Substitute
Now we need to substitute back \( t = \sin^{-1}(\sqrt{x}) \):
\[ \int \sin^{-1}(\sqrt{x}) \, dx = -\frac{1}{2} \sin^{-1}(\sqrt{x}) \cos(2\sin^{-1}(\sqrt{x})) + \frac{1}{4} \sin(2\sin^{-1}(\sqrt{x})) + C \]

### Step 6: Simplifying the Expression
Using the identities:
- \( \cos(2\sin^{-1}(\sqrt{x})) = 1 - 2\sin^2(\sin^{-1}(\sqrt{x})) = 1 - 2x \)
- \( \sin(2\sin^{-1}(\sqrt{x})) = 2\sin(\sin^{-1}(\sqrt{x}))\cos(\sin^{-1}(\sqrt{x})) = 2\sqrt{x}\sqrt{1-x} \)

We can rewrite the integral as:
\[ \int \sin^{-1}(\sqrt{x}) \, dx = -\frac{1}{2} \sin^{-1}(\sqrt{x})(1 - 2x) + \frac{1}{4}(2\sqrt{x}\sqrt{1-x}) + C \]

### Final Answer
Thus, the final result is:
\[ \int \sin^{-1}(\sqrt{x}) \, dx = -\frac{1}{2} \sin^{-1}(\sqrt{x})(1 - 2x) + \frac{1}{2}\sqrt{x(1-x)} + C \]