Let `f (x)` be function defined on `[0,1] ` such that `f (1)=0` and for any `a in (0,1], int _(0)^(a) f (x) dx - int _(a)^(1) f (x) dx =2 f (a) +3a +b` where b is constant.
b=

Options

A. `(3)/(2e)-3`
B. `(3)/(2e)-(3)/(2)`
C. `(3)/(2e) +3`
D. `(3)/(2e)+ 3/2`

Answer

To solve the problem, we will follow a systematic approach step by step.

### Step 1: Understanding the Given Equation
We are given the equation:
\[ \int_0^a f(x) \, dx - \int_a^1 f(x) \, dx = 2f(a) + 3a + b \]
for \( a \in (0, 1] \) and \( f(1) = 0 \).

### Step 2: Differentiating with Respect to \( a \)
To analyze the equation further, we differentiate both sides with respect to \( a \):
\[ \frac{d}{da} \left( \int_0^a f(x) \, dx - \int_a^1 f(x) \, dx \right) = \frac{d}{da} \left( 2f(a) + 3a + b \right) \]

Using the Fundamental Theorem of Calculus and the Leibniz rule, we get:
\[ f(a) - (-f(a)) = 2f'(a) + 3 \]
This simplifies to:
\[ 2f(a) = 2f'(a) + 3 \]

### Step 3: Rearranging the Equation
We can rearrange the equation to isolate \( f'(a) \):
\[ 2f'(a) = 2f(a) - 3 \]
Dividing through by 2 gives:
\[ f'(a) = f(a) - \frac{3}{2} \]

### Step 4: Solving the Differential Equation
This is a first-order linear differential equation. We can solve it using the integrating factor method. The integrating factor \( e^{-\int 1 \, da} = e^{-a} \).

Multiplying both sides by the integrating factor:
\[ e^{-a} f'(a) - \frac{3}{2} e^{-a} = 0 \]
This can be rewritten as:
\[ \frac{d}{da} \left( e^{-a} f(a) \right) = \frac{3}{2} e^{-a} \]

### Step 5: Integrating Both Sides
Integrating both sides gives:
\[ e^{-a} f(a) = -\frac{3}{2} e^{-a} + C \]
Multiplying through by \( e^{a} \):
\[ f(a) = -\frac{3}{2} + Ce^{a} \]

### Step 6: Applying the Boundary Condition
We know that \( f(1) = 0 \):
\[ 0 = -\frac{3}{2} + Ce^{1} \]
Solving for \( C \):
\[ C = \frac{3}{2e} \]

### Step 7: Final Expression for \( f(a) \)
Substituting \( C \) back into the equation for \( f(a) \):
\[ f(a) = -\frac{3}{2} + \frac{3}{2e} e^{a} \]
This simplifies to:
\[ f(a) = \frac{3}{2} \left( e^{a} - 1 \right) \]

### Step 8: Finding the Value of \( b \)
Now, we need to find the value of \( b \). We can substitute \( a = 1 \) back into the original equation:
\[ \int_0^1 f(x) \, dx - 0 = 2f(1) + 3(1) + b \]
Since \( f(1) = 0 \):
\[ \int_0^1 f(x) \, dx = 3 + b \]

Calculating \( \int_0^1 f(x) \, dx \):
\[ \int_0^1 \frac{3}{2} (e^{x} - 1) \, dx = \frac{3}{2} \left( \left[ e^{x} \right]_0^1 - \left[ x \right]_0^1 \right) = \frac{3}{2} \left( e - 1 - 1 \right) = \frac{3}{2} (e - 2) \]

Setting this equal to \( 3 + b \):
\[ \frac{3}{2} (e - 2) = 3 + b \]

### Step 9: Solving for \( b \)
Rearranging gives:
\[ b = \frac{3}{2} (e - 2) - 3 = \frac{3e - 6 - 6}{2} = \frac{3e - 12}{2} \]

Thus, the value of \( b \) is:
\[ b = \frac{3e - 12}{2} \]

### Summary of Steps
1. Understand the equation and differentiate with respect to \( a \).
2. Rearrange the resulting equation.
3. Solve the first-order linear differential equation.
4. Apply boundary conditions to find constants.
5. Substitute back to find \( b \).