`int(1)/(x^(1//2)(1+x^(2))^(5//4))dx` is equal to

Options

A. `(-2sqrt(x))/(4sqrt(1+x^(2)))+C`
B. `(2sqrt(x))/(4sqrt(1+x^(2)))+C`
C. `(-sqrt(x))/(4sqrt(1+x^(2)))+C`
D. `(sqrt(x))/(4sqrt(1+x^(2)))+C`

Answer

To solve the integral

\[ \int \frac{1}{x^{1/2}(1+x^2)^{5/4}} \, dx, \]

we will follow a systematic approach.

### Step 1: Rewrite the Integral
We start with the integral:

\[ \int \frac{1}{x^{1/2}(1+x^2)^{5/4}} \, dx. \]

### Step 2: Factor out \(x^2\) from the Denominator
Notice that we can express \(1 + x^2\) in a more manageable form. We can factor \(x^2\) out of the term \(1 + x^2\):

\[ 1 + x^2 = x^2\left(\frac{1}{x^2} + 1\right) = x^2\left(1 + \frac{1}{x^2}\right). \]

Thus, we can rewrite the integral as:

\[ \int \frac{1}{x^{1/2} \cdot (x^2)^{5/4} \cdot \left(1 + \frac{1}{x^2}\right)^{5/4}} \, dx. \]

### Step 3: Simplify the Integral
This simplifies to:

\[ \int \frac{1}{x^{1/2} \cdot x^{5/2} \cdot \left(1 + \frac{1}{x^2}\right)^{5/4}} \, dx = \int \frac{1}{x^3 \cdot \left(1 + \frac{1}{x^2}\right)^{5/4}} \, dx. \]

### Step 4: Substitute \(t = 1 + \frac{1}{x^2}\)
Let’s make a substitution to simplify the integral further. Set:

\[ t = 1 + \frac{1}{x^2} \implies \frac{dt}{dx} = -\frac{2}{x^3} \implies dx = -\frac{x^3}{2} dt. \]

### Step 5: Substitute into the Integral
Now we substitute \(dx\) and \(t\) into the integral:

\[ \int \frac{-\frac{x^3}{2}}{x^3 \cdot t^{5/4}} \, dt = -\frac{1}{2} \int t^{-5/4} \, dt. \]

### Step 6: Integrate
Now we integrate:

\[ -\frac{1}{2} \int t^{-5/4} \, dt = -\frac{1}{2} \cdot \left(\frac{t^{-1/4}}{-1/4}\right) = 2 t^{-1/4} + C. \]

### Step 7: Substitute Back for \(t\)
Now we substitute back for \(t\):

\[ 2 \left(1 + \frac{1}{x^2}\right)^{-1/4} + C. \]

### Step 8: Final Answer
Thus, the final answer is:

\[ \frac{2}{(1 + \frac{1}{x^2})^{1/4}} + C = \frac{2 \sqrt{x^2}}{(x^2 + 1)^{1/4}} + C = \frac{2x}{(x^2 + 1)^{1/4}} + C. \]