`int(sin^(-1)x+cos^(-1)x)dx=`

Options

A. `(pix)/(2)+c`
B. `(pix)/(2)+x+c`
C. `x(sin^(-1)x-cos^(-1)x)+c`
D. `x(sin^(-1)x+cos^(-1)x)+c`

Answer

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

### Step 1: Recognize the relationship between the inverse trigonometric functions
We know that:
\[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \]
This is a fundamental identity for inverse sine and cosine functions.

### Step 2: Substitute the identity into the integral
Using the identity from Step 1, we can rewrite the integral:
\[ \int (\sin^{-1} x + \cos^{-1} x) \, dx = \int \frac{\pi}{2} \, dx \]

### Step 3: Integrate the constant
Now, we can integrate the constant \( \frac{\pi}{2} \):
\[ \int \frac{\pi}{2} \, dx = \frac{\pi}{2} x + C \]
where \( C \) is the constant of integration.

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