If the integral `I=inte^(sinx)(cosx.x^(2)+2x)dx=e^(f(x))g(x)+C` (where, C is the constant of integration), then the number of solution(s) of `f(x)=g(x)` is/are

Options

A. 0
B. 2
C. 4
D. 6

Answer

To solve the problem, we need to analyze the integral given and determine the functions \( f(x) \) and \( g(x) \) such that \( f(x) = g(x) \).

### Step-by-Step Solution:

1. **Identify the Integral**:
We have the integral:
\[ I = \int e^{\sin x} \left( \cos x \cdot (x^2 + 2x) \right) dx \]

2. **Recognize the Form**:
The integral can be expressed in the form:
\[ \int e^{f(x)} g'(x) \, dx = e^{f(x)} g(x) + C \]
where \( f'(x) = \cos x \) and \( g'(x) = x^2 + 2x \).

3. **Determine \( f(x) \) and \( g(x) \)**:
- From \( f'(x) = \cos x \), we can integrate to find \( f(x) \):
\[ f(x) = \sin x + C_1 \]
- From \( g'(x) = x^2 + 2x \), we integrate to find \( g(x) \):
\[ g(x) = \frac{x^3}{3} + x^2 + C_2 \]

4. **Set Up the Equation**:
We need to find the number of solutions to the equation:
\[ f(x) = g(x) \]
This translates to:
\[ \sin x = \frac{x^3}{3} + x^2 + C \]
where \( C \) is a constant that can be absorbed into \( C_1 \) or \( C_2 \).

5. **Analyze the Functions**:
- The left-hand side \( \sin x \) oscillates between -1 and 1.
- The right-hand side \( \frac{x^3}{3} + x^2 + C \) is a cubic polynomial that increases without bound as \( x \) increases.

6. **Graphical Interpretation**:
- The graph of \( \sin x \) will intersect the graph of \( \frac{x^3}{3} + x^2 + C \).
- Since \( \sin x \) oscillates, we can expect it to intersect the cubic polynomial at various points.

7. **Finding Intersections**:
- For small values of \( x \), \( \sin x \) will cross the cubic polynomial.
- As \( x \) increases, the cubic polynomial will eventually exceed the maximum value of \( \sin x \) (which is 1).

8. **Count the Solutions**:
- The cubic polynomial will intersect the sine function twice before it surpasses the maximum of 1.
- Therefore, the number of solutions to \( f(x) = g(x) \) is **2**.

### Conclusion:
The number of solutions to the equation \( f(x) = g(x) \) is **2**.