`int cos^(-1) (sinx) dx`

Answer

To solve the integral \( \int \cos^{-1}(\sin x) \, dx \), we will follow the steps outlined in the video solution.

### Step-by-step Solution:

1. **Rewrite the Integral**:
We start with the integral:
\[ I = \int \cos^{-1}(\sin x) \, dx \]
We know that \( \sin x = \cos\left(\frac{\pi}{2} - x\right) \). Therefore, we can rewrite the integral as:
\[ I = \int \cos^{-1}\left(\cos\left(\frac{\pi}{2} - x\right)\right) \, dx \]

2. **Apply the Inverse Cosine Identity**:
The property of the inverse cosine function states that \( \cos^{-1}(\cos(a)) = a \) when \( a \) is in the range of \( \cos^{-1} \). Thus, we have:
\[ I = \int \left(\frac{\pi}{2} - x\right) \, dx \]

3. **Separate the Integral**:
We can separate the integral into two parts:
\[ I = \int \frac{\pi}{2} \, dx - \int x \, dx \]

4. **Integrate Each Part**:
Now we integrate each term separately:
- The integral of a constant:
\[ \int \frac{\pi}{2} \, dx = \frac{\pi}{2} x \]
- The integral of \( x \):
\[ \int x \, dx = \frac{x^2}{2} \]

5. **Combine the Results**:
Putting it all together, we have:
\[ I = \frac{\pi}{2} x - \frac{x^2}{2} + C \]
where \( C \) is the constant of integration.

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