if `f(x)` is a differential function such that `f(x)=int_(0)^(x)(1+2xf(t))dt&f(1)=e`, then
Q. Identify the correct statements (s):

Options

A. `f(x)` is an even function
B. `f(x)` is an odd function
C. `f(x)` is always increasing function
D. `f(x)` is always decreasing function

Answer

To solve the problem step by step, we start with the given information about the function \( f(x) \):

1. **Given Function**:
\[ f(x) = \int_0^x (1 + 2x f(t)) dt \]
and \( f(1) = e \).

2. **Rearranging the Integral**:
We can rewrite the function as:
\[ f(x) = \int_0^x 1 \, dt + \int_0^x 2x f(t) \, dt \]
The first integral evaluates to \( x \):
\[ f(x) = x + 2x \int_0^x f(t) \, dt \]

3. **Isolate the Integral**:
Rearranging gives:
\[ f(x) - x = 2x \int_0^x f(t) \, dt \]
Dividing both sides by \( 2x \) (assuming \( x \neq 0 \)):
\[ \frac{f(x) - x}{2x} = \int_0^x f(t) \, dt \]

4. **Differentiate Both Sides**:
Now, we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}\left(\frac{f(x) - x}{2x}\right) = \frac{d}{dx}\left(\int_0^x f(t) \, dt\right) \]
Using the quotient rule on the left side:
\[ \frac{(f'(x) - 1) \cdot 2x - (f(x) - x) \cdot 2}{(2x)^2} = f(x) \]
Simplifying gives:
\[ (f'(x) - 1) \cdot x - (f(x) - x) = x^2 f(x) \]

5. **Rearranging the Equation**:
This leads to:
\[ x f'(x) - f(x) = x^2 f(x) + x \]
Rearranging yields:
\[ x f'(x) = (x^2 + 1) f(x) + x \]

6. **Expressing \( f'(x) \)**:
Thus, we can express \( f'(x) \) as:
\[ f'(x) = \frac{(x^2 + 1) f(x) + x}{x} \]

7. **Finding the General Solution**:
To solve this differential equation, we can separate variables:
\[ \frac{f'(x)}{f(x)} = \frac{x^2 + 1}{x} + \frac{1}{x} \]
Integrating both sides gives:
\[ \ln |f(x)| = \frac{x^2}{2} + \ln |x| + C \]
Therefore:
\[ f(x) = k \cdot x \cdot e^{\frac{x^2}{2}} \]
where \( k = e^C \).

8. **Using the Initial Condition**:
We know \( f(1) = e \):
\[ f(1) = k \cdot 1 \cdot e^{\frac{1^2}{2}} = k \cdot e^{\frac{1}{2}} = e \]
Solving for \( k \):
\[ k = e^{1 - \frac{1}{2}} = e^{\frac{1}{2}} = \sqrt{e} \]

9. **Final Form of \( f(x) \)**:
Thus, we have:
\[ f(x) = \sqrt{e} \cdot x \cdot e^{\frac{x^2}{2}} = x \cdot e^{\frac{x^2}{2} + \frac{1}{2}} \]

10. **Identifying the Properties of \( f(x) \)**:
- Since \( e^{\frac{x^2}{2}} \) is always positive and increasing, \( f(x) \) is an increasing function.
- To check if \( f(-x) = -f(x) \), we find:
\[ f(-x) = -x \cdot e^{\frac{x^2}{2} + \frac{1}{2}} = -f(x) \]
Thus, \( f(x) \) is an odd function.

### Conclusion:
The correct statements about \( f(x) \) are:
- \( f(x) \) is an increasing function.
- \( f(x) \) is an odd function.