`int(x^(7//2))/(sqrt(x^(9)))dx` equal to

Options

A. `sin^(-1)sqrt(x^(2)+9)+c`
B. `(1)/(2)ln((x^(9)+1)/(x^(9)-1))+c`
C. `(2)/(9)ln(x^(9//2))+c`
D. none of these

Answer

To solve the integral \( \int \frac{x^{7/2}}{\sqrt{x^9}} \, dx \), we can follow these steps:

### Step 1: Simplify the integrand
First, we simplify the expression inside the integral:
\[ \sqrt{x^9} = x^{9/2} \]
Thus, we can rewrite the integral as:
\[ \int \frac{x^{7/2}}{x^{9/2}} \, dx \]

### Step 2: Combine the powers of \(x\)
Now, we can combine the powers of \(x\):
\[ \frac{x^{7/2}}{x^{9/2}} = x^{7/2 - 9/2} = x^{-1} \]
So the integral becomes:
\[ \int x^{-1} \, dx \]

### Step 3: Integrate
The integral of \(x^{-1}\) is:
\[ \int x^{-1} \, dx = \log |x| + C \]
where \(C\) is the constant of integration.

### Step 4: Final expression
Thus, the final result of the integral is:
\[ \log |x| + C \]