Integration of f (x) `= sqrt(1+x^(2))` with respect to `x^(2)`, is

Options

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

Explanation

To solve the integral of \( f(x) = \sqrt{1 + x^2} \) with respect to \( x^2 \), we will follow these steps:

### Step 1: Rewrite the Integral
We want to find:
\[ \int \sqrt{1 + x^2} \, d(x^2) \]
Since \( d(x^2) = 2x \, dx \), we can rewrite the integral as:
\[ \int \sqrt{1 + x^2} \cdot 2x \, dx \]

### Step 2: Simplify the Integral
Now we can factor out the constant:
\[ 2 \int x \sqrt{1 + x^2} \, dx \]

### Step 3: Use Substitution
Let’s use the substitution:
\[ t = 1 + x^2 \implies dt = 2x \, dx \]
This means:
\[ x \, dx = \frac{dt}{2} \]
Substituting in the integral, we have:
\[ 2 \int x \sqrt{1 + x^2} \, dx = 2 \int x \sqrt{t} \cdot \frac{dt}{2} = \int \sqrt{t} \, dt \]

### Step 4: Integrate
Now we can integrate:
\[ \int \sqrt{t} \, dt = \int t^{1/2} \, dt = \frac{t^{3/2}}{3/2} + C = \frac{2}{3} t^{3/2} + C \]

### Step 5: Substitute Back
Substituting back \( t = 1 + x^2 \):
\[ \frac{2}{3} (1 + x^2)^{3/2} + C \]

### Final Answer
Thus, the integral of \( f(x) = \sqrt{1 + x^2} \) with respect to \( x^2 \) is:
\[ \frac{2}{3} (1 + x^2)^{3/2} + C \]