Evaluate the following integrals:
`int(sin^(-1) sqrt(x)+cos^(-1)sqrt(x))dx`

Answer

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

### Step 1: Simplify the expression
We start by using the identity that relates the inverse sine and inverse cosine functions:
\[ \sin^{-1}(y) + \cos^{-1}(y) = \frac{\pi}{2} \]
for any \( y \) in the range \([-1, 1]\). In our case, we have:
\[ \sin^{-1}(\sqrt{x}) + \cos^{-1}(\sqrt{x}) = \frac{\pi}{2} \]

### Step 2: Substitute into the integral
Now we can substitute this result into the integral:
\[ \int \left( \sin^{-1}(\sqrt{x}) + \cos^{-1}(\sqrt{x}) \right) \, dx = \int \frac{\pi}{2} \, dx \]

### Step 3: Evaluate the integral
The integral of a constant is straightforward:
\[ \int \frac{\pi}{2} \, dx = \frac{\pi}{2} x + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the evaluated integral is:
\[ \int \left( \sin^{-1}(\sqrt{x}) + \cos^{-1}(\sqrt{x}) \right) \, dx = \frac{\pi}{2} x + C \]

---