`int (x^5)/(sqrt((1 + x^2))) dx = `

Options

A. `1/15 sqrt((1 + x^2)) [3x^4 + 4x^2 + 8]`
B. `1/15 sqrt((1 + x^2)) [3x^4 - 4x^2 + 8]`
C. `sqrt((1 + x^2)) [3x^4 + 4x^2 + 8]`
D. none

Answer

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

### Step 1: Break down the numerator
We can rewrite \( x^5 \) as \( x^4 \cdot x \). This allows us to separate the powers of \( x \):
\[ \int \frac{x^5}{\sqrt{1 + x^2}} \, dx = \int \frac{x^4 \cdot x}{\sqrt{1 + x^2}} \, dx \]

**Hint**: Consider breaking down the powers of \( x \) to simplify the integration process.

### Step 2: Substitution
Let \( t = 1 + x^2 \). Then, differentiate \( t \) with respect to \( x \):
\[ dt = 2x \, dx \quad \Rightarrow \quad dx = \frac{dt}{2x} \]
Also, from our substitution, we have \( x^2 = t - 1 \), which gives \( x^4 = (t - 1)^2 \).

**Hint**: Use substitution to simplify the integral, especially when dealing with square roots.

### Step 3: Substitute into the integral
Substituting \( x^4 \) and \( dx \) into the integral, we get:
\[ \int \frac{(t - 1)^2 \cdot x}{\sqrt{t}} \cdot \frac{dt}{2x} = \frac{1}{2} \int \frac{(t - 1)^2}{\sqrt{t}} \, dt \]

**Hint**: Simplifying the integral after substitution can make it easier to integrate.

### Step 4: Expand the integrand
Now, expand \( (t - 1)^2 \):
\[ (t - 1)^2 = t^2 - 2t + 1 \]
Thus, the integral becomes:
\[ \frac{1}{2} \int \frac{t^2 - 2t + 1}{\sqrt{t}} \, dt = \frac{1}{2} \int \left( t^{3/2} - 2t^{1/2} + t^{-1/2} \right) \, dt \]

**Hint**: Expanding the integrand can help in integrating term by term.

### Step 5: Integrate term by term
Now we can integrate each term separately:
\[ \frac{1}{2} \left( \frac{2}{5} t^{5/2} - 2 \cdot \frac{2}{3} t^{3/2} + 2t^{1/2} \right) + C \]
This simplifies to:
\[ \frac{1}{5} t^{5/2} - \frac{2}{3} t^{3/2} + t^{1/2} + C \]

**Hint**: Use the power rule for integration to handle each term.

### Step 6: Substitute back for \( t \)
Now substitute back \( t = 1 + x^2 \):
\[ = \frac{1}{5} (1 + x^2)^{5/2} - \frac{2}{3} (1 + x^2)^{3/2} + (1 + x^2)^{1/2} + C \]

**Hint**: Always remember to revert back to the original variable after integration.

### Final Result
Thus, the final answer is:
\[ \int \frac{x^5}{\sqrt{1 + x^2}} \, dx = \frac{1}{5} (1 + x^2)^{5/2} - \frac{2}{3} (1 + x^2)^{3/2} + \sqrt{1 + x^2} + C \]