If the antiderivative of `x^3/sqrt(1+2x^2)` which passes through `(1, 2)` is `1/m(1+2x^2)^(1/2)(x^2-1)+2,` then the value of `m` is

Options

A. 1
B. 3
C. 5
D. 6

Answer

To solve the problem, we need to find the value of \( m \) such that the antiderivative of \( \frac{x^3}{\sqrt{1 + 2x^2}} \) that passes through the point \( (1, 2) \) is given by the expression:

\[ \frac{1}{m} (1 + 2x^2)^{1/2} (x^2 - 1) + 2 \]

### Step 1: Set up the integral
We start with the integral:

\[ y = \int \frac{x^3}{\sqrt{1 + 2x^2}} \, dx \]

### Step 2: Use substitution
Let \( t^2 = 1 + 2x^2 \). Then, differentiating both sides gives:

\[ 2t \frac{dt}{dx} = 4x \implies \frac{dt}{dx} = \frac{2x}{t} \implies dt = \frac{2x}{t} \, dx \implies dx = \frac{t}{2x} \, dt \]

### Step 3: Express \( x^3 \) in terms of \( t \)
From \( t^2 = 1 + 2x^2 \), we can express \( x^2 \) as:

\[ x^2 = \frac{t^2 - 1}{2} \]

Thus, \( x^3 = x \cdot x^2 = x \cdot \frac{t^2 - 1}{2} \).

### Step 4: Rewrite the integral
Substituting \( x^3 \) and \( dx \) into the integral gives:

\[ y = \int \frac{\left( \frac{t^2 - 1}{2} \right) x \cdot \frac{t}{2x}}{\sqrt{t^2}} \, dt \]

The \( x \) cancels out, and we have:

\[ y = \frac{1}{4} \int (t^2 - 1) \, dt \]

### Step 5: Integrate
Now we integrate:

\[ y = \frac{1}{4} \left( \frac{t^3}{3} - t \right) + C \]

### Step 6: Substitute back for \( t \)
Substituting back \( t = \sqrt{1 + 2x^2} \):

\[ y = \frac{1}{4} \left( \frac{(1 + 2x^2)^{3/2}}{3} - \sqrt{1 + 2x^2} \right) + C \]

### Step 7: Simplify
This simplifies to:

\[ y = \frac{1}{12} (1 + 2x^2)^{3/2} - \frac{1}{4} \sqrt{1 + 2x^2} + C \]

### Step 8: Find the constant \( C \)
We know that the function passes through the point \( (1, 2) \):

\[ 2 = \frac{1}{12} (1 + 2 \cdot 1^2)^{3/2} - \frac{1}{4} \sqrt{1 + 2 \cdot 1^2} + C \]

Calculating \( (1 + 2 \cdot 1^2) = 3 \):

\[ 2 = \frac{1}{12} (3)^{3/2} - \frac{1}{4} \cdot \sqrt{3} + C \]

Calculating \( (3)^{3/2} = 3\sqrt{3} \):

\[ 2 = \frac{3\sqrt{3}}{12} - \frac{\sqrt{3}}{4} + C \]

Finding a common denominator (12):

\[ 2 = \frac{3\sqrt{3}}{12} - \frac{3\sqrt{3}}{12} + C \]

Thus, \( C = 2 \).

### Step 9: Final expression
Now we have:

\[ y = \frac{1}{12} (1 + 2x^2)^{3/2} - \frac{1}{4} \sqrt{1 + 2x^2} + 2 \]

### Step 10: Compare with given expression
The given expression is:

\[ \frac{1}{m} (1 + 2x^2)^{1/2} (x^2 - 1) + 2 \]

By comparing coefficients, we find that:

\[ \frac{1}{m} = \frac{1}{6} \implies m = 6 \]

### Conclusion
Thus, the value of \( m \) is:

\[ \boxed{6} \]