Choose the correct answerIf `f(x)=int_0^x tsint dt`, then `f^(prime)(x)`is(A) `cos x + x sin x` (B) `x sin x` (C) `x cos x` (D) `sin x+ x cos x`

Options

A. (A) `cos x + x sin x`
B. (B) `x sin x`
C. (C) `x cos x`
D. (D) `sin x+ x cos x`

Explanation

To solve the problem, we need to find the derivative of the function defined by the integral \( f(x) = \int_0^x t \sin t \, dt \).

### Step-by-step Solution:

1. **Identify the Function**:
We have the function defined as:
\[ f(x) = \int_0^x t \sin t \, dt \]

2. **Apply the Fundamental Theorem of Calculus**:
According to the Fundamental Theorem of Calculus, if \( F(x) = \int_a^x f(t) \, dt \), then \( F'(x) = f(x) \). In our case, we can differentiate \( f(x) \):
\[ f'(x) = x \sin x \]

3. **Conclusion**:
Therefore, the derivative \( f'(x) \) is:
\[ f'(x) = x \sin x \]

4. **Choose the Correct Answer**:
From the options provided:
- (A) \( \cos x + x \sin x \)
- (B) \( x \sin x \)
- (C) \( x \cos x \)
- (D) \( \sin x + x \cos x \)

The correct answer is **(B) \( x \sin x \)**.