`intsqrt(x-3)(sin^(-1)(Inx)+cos^(-1)(Inx))dx` is equal to

Options

A. `(pi)/(3)(x-3)^(3//2)+C`
B. 0
C. 1
D. none of these

Explanation

To solve the integral \(\int \sqrt{x-3} \left( \sin^{-1}(\ln x) + \cos^{-1}(\ln x) \right) dx\), we can follow these steps:

### Step 1: Simplify the expression inside the integral
We know that:
\[ \sin^{-1}(\theta) + \cos^{-1}(\theta) = \frac{\pi}{2} \]
for any \(\theta\). In our case, \(\theta = \ln x\). Therefore, we can rewrite the integral as:
\[ \int \sqrt{x-3} \left( \sin^{-1}(\ln x) + \cos^{-1}(\ln x) \right) dx = \int \sqrt{x-3} \cdot \frac{\pi}{2} \, dx \]

### Step 2: Factor out the constant
Since \(\frac{\pi}{2}\) is a constant, we can factor it out of the integral:
\[ = \frac{\pi}{2} \int \sqrt{x-3} \, dx \]

### Step 3: Integrate \(\sqrt{x-3}\)
To integrate \(\sqrt{x-3}\), we can use the power rule for integration. We rewrite \(\sqrt{x-3}\) as \((x-3)^{1/2}\):
\[ \int (x-3)^{1/2} \, dx \]
Using the formula for integration:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
where \(n = \frac{1}{2}\):
\[ = \frac{(x-3)^{3/2}}{\frac{3}{2}} + C = \frac{2}{3} (x-3)^{3/2} + C \]

### Step 4: Combine the results
Now, substituting back into our integral:
\[ \frac{\pi}{2} \int \sqrt{x-3} \, dx = \frac{\pi}{2} \cdot \left( \frac{2}{3} (x-3)^{3/2} + C \right) \]
This simplifies to:
\[ = \frac{\pi}{3} (x-3)^{3/2} + C \]

### Final Result
Thus, the final result of the integral is:
\[ \int \sqrt{x-3} \left( \sin^{-1}(\ln x) + \cos^{-1}(\ln x) \right) dx = \frac{\pi}{3} (x-3)^{3/2} + C \]