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

Options

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

Answer

To solve the integral \( \int \frac{1}{x^{1/2} (1+x^2)^{5/4}} \, dx \), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{1}{x^{1/2} (1+x^2)^{5/4}} \, dx \]

### Step 2: Simplify the Denominator
We can rewrite the denominator by factoring out \( x^2 \) from \( (1+x^2)^{5/4} \):
\[ I = \int \frac{1}{x^{1/2} \cdot (1+x^2)^{5/4}} \, dx = \int \frac{1}{x^{1/2} \cdot (x^2(1+\frac{1}{x^2}))^{5/4}} \, dx \]
This gives us:
\[ I = \int \frac{1}{x^{1/2} \cdot x^{5/2} \cdot (1+\frac{1}{x^2})^{5/4}} \, dx = \int \frac{1}{x^3 (1+\frac{1}{x^2})^{5/4}} \, dx \]

### Step 3: Substitute \( t = 1 + x^2 \)
Now, we will use the substitution \( t = 1 + x^2 \). Then, \( dt = 2x \, dx \) or \( dx = \frac{dt}{2\sqrt{t-1}} \). Also, \( x^2 = t - 1 \) and \( x = \sqrt{t-1} \).

Substituting these into the integral gives:
\[ I = \int \frac{1}{(t-1)^{3/2} (t)^{5/4}} \cdot \frac{dt}{2\sqrt{t-1}} \]

### Step 4: Simplify the Integral
This simplifies to:
\[ I = \frac{1}{2} \int \frac{1}{(t-1)^2 t^{5/4}} \, dt \]

### Step 5: Integrate
Now we can integrate:
Using the formula for integration \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \):
\[ I = \frac{1}{2} \left( -\frac{4}{t^{1/4}} + C \right) \]

### Step 6: Substitute Back
Now we substitute back \( t = 1 + x^2 \):
\[ I = -\frac{2}{(1+x^2)^{1/4}} + C \]

### Final Result
Thus, the final result of the integral is:
\[ I = \frac{2\sqrt{x}}{\sqrt[4]{1+x^2}} + C \]