`int(1)/((a^(2)+x^(2))^(3//2))dx` is equal to

Options

A. `(x)/(a^(2)sqrt(a^(2)+x^(2)))+C`
B. `(x)/((a^(2)+x^(2))^(3//3))+C`
C. `(1)/(a^(2)sqrt(a^(2)+x^(2)))+C`
D. none of these

Answer

To solve the integral \( I = \int \frac{1}{(a^2 + x^2)^{3/2}} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{1}{(a^2 + x^2)^{3/2}} \, dx \]

### Step 2: Use a Substitution
We will use the substitution \( t = \frac{x^2}{a^2 + x^2} \). This implies:
\[ 1 + \frac{a^2}{x^2} = \frac{a^2 + x^2}{x^2} \implies \frac{a^2}{x^2} = \frac{1}{t} - 1 \]
Differentiating \( t \) gives us:
\[ dt = \frac{2x}{(a^2 + x^2)^2} \, dx \implies dx = \frac{(a^2 + x^2)^2}{2x} \, dt \]

### Step 3: Substitute into the Integral
Now substituting \( dx \) in terms of \( dt \):
\[ I = \int \frac{(a^2 + x^2)^2}{2x} \cdot \frac{1}{(a^2 + x^2)^{3/2}} \, dt \]
This simplifies to:
\[ I = \int \frac{(a^2 + x^2)^{1/2}}{2x} \, dt \]

### Step 4: Simplify the Integral
Now we can simplify further. Notice that:
\[ \frac{(a^2 + x^2)^{1/2}}{x} = \sqrt{\frac{a^2 + x^2}{x^2}} = \sqrt{\frac{a^2}{x^2} + 1} = \sqrt{1 + \frac{a^2}{x^2}} = \sqrt{1 + t} \]
Thus, we can rewrite the integral as:
\[ I = \frac{1}{2} \int \sqrt{1 + t} \, dt \]

### Step 5: Integrate
The integral of \( \sqrt{1 + t} \) is:
\[ \int \sqrt{1 + t} \, dt = \frac{2}{3}(1 + t)^{3/2} + C \]
So, substituting back, we have:
\[ I = \frac{1}{2} \cdot \frac{2}{3} (1 + t)^{3/2} + C = \frac{1}{3} (1 + t)^{3/2} + C \]

### Step 6: Substitute Back for \( t \)
Recall that \( t = \frac{x^2}{a^2 + x^2} \):
\[ I = \frac{1}{3} \left(1 + \frac{x^2}{a^2 + x^2}\right)^{3/2} + C = \frac{1}{3} \left(\frac{a^2 + 2x^2}{a^2 + x^2}\right)^{3/2} + C \]

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