`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
We start by rewriting the integrand. The square root in the denominator can be expressed as a power of \( x \):
\[ \sqrt{x^9} = x^{9/2} \]
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 simplify the fraction:
\[ \frac{x^{7/2}}{x^{9/2}} = x^{7/2 - 9/2} = x^{-1} \]
So the integral becomes:
\[ \int x^{-1} \, dx \]

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

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

### Summary of Steps:
1. Simplify the integrand by rewriting the square root.
2. Combine the powers of \( x \).
3. Integrate the resulting expression.
4. Write the final result with the constant of integration.