If ` int _(0)^(x^(2)) f(t) dt = x cos pix` , then the value of f(4) is

Options

A. 1
B. `1/4`
C. `-1`
D. `(-1)/4`

Explanation

To solve the problem, we need to find the value of \( f(4) \) given that

\[ \int_{0}^{x^2} f(t) \, dt = x \cos(\pi x). \]

### Step 1: Differentiate both sides with respect to \( x \)

Using the Fundamental Theorem of Calculus and the chain rule, we differentiate the left-hand side:

\[ \frac{d}{dx} \left( \int_{0}^{x^2} f(t) \, dt \right) = f(x^2) \cdot \frac{d}{dx}(x^2) = f(x^2) \cdot 2x. \]

Now, we differentiate the right-hand side:

\[ \frac{d}{dx}(x \cos(\pi x)) = \cos(\pi x) + x \cdot \frac{d}{dx}(\cos(\pi x)). \]

Using the chain rule, we find:

\[ \frac{d}{dx}(\cos(\pi x)) = -\pi \sin(\pi x). \]

Thus, the differentiation of the right-hand side becomes:

\[ \cos(\pi x) - \pi x \sin(\pi x). \]

### Step 2: Set the derivatives equal to each other

Now we set the derivatives from both sides equal to each other:

\[ f(x^2) \cdot 2x = \cos(\pi x) - \pi x \sin(\pi x). \]

### Step 3: Solve for \( f(x^2) \)

To isolate \( f(x^2) \), we divide both sides by \( 2x \):

\[ f(x^2) = \frac{\cos(\pi x) - \pi x \sin(\pi x)}{2x}. \]

### Step 4: Find \( f(4) \)

To find \( f(4) \), we need to set \( x^2 = 4 \) which gives \( x = 2 \):

\[ f(4) = \frac{\cos(2\pi) - \pi \cdot 2 \sin(2\pi)}{2 \cdot 2}. \]

### Step 5: Evaluate \( \cos(2\pi) \) and \( \sin(2\pi) \)

We know:

\[ \cos(2\pi) = 1 \quad \text{and} \quad \sin(2\pi) = 0. \]

### Step 6: Substitute these values into the equation

Substituting these values into our expression for \( f(4) \):

\[ f(4) = \frac{1 - \pi \cdot 2 \cdot 0}{4} = \frac{1}{4}. \]

### Final Answer

Thus, the value of \( f(4) \) is

\[ \boxed{\frac{1}{4}}. \]