If `inte^(x)[f(x) -f'(x)]dx =Psi(x)`, then `inte^(x)f(x)` dx is

Options

A. `Psi(x)+e^xf(x)`
B. `Psi(x)-e^(x)f(x)`
C. `1/2[Psi(x)+e^x f(x)]`
D. `1/2[Psi(x)+e^x f'(x)]`

Answer

To solve the problem, we start with the given equation:

\[ \int e^x [f(x) - f'(x)] \, dx = \Psi(x) \]

We want to find the integral:

\[ \int e^x f(x) \, dx \]

### Step 1: Expand the given integral

We can expand the left-hand side of the given equation:

\[ \int e^x f(x) \, dx - \int e^x f'(x) \, dx = \Psi(x) \]

### Step 2: Use Integration by Parts on the second term

We will focus on the term \(\int e^x f'(x) \, dx\). We can apply integration by parts here. Recall the formula for integration by parts:

\[ \int u \, dv = uv - \int v \, du \]

Let:

- \(u = f'(x)\) (thus, \(du = f''(x) \, dx\))
- \(dv = e^x \, dx\) (thus, \(v = e^x\))

Using integration by parts, we have:

\[ \int e^x f'(x) \, dx = e^x f'(x) - \int e^x f''(x) \, dx \]

### Step 3: Substitute back into the equation

Now, substituting this back into our expanded equation:

\[ \int e^x f(x) \, dx - \left( e^x f'(x) - \int e^x f''(x) \, dx \right) = \Psi(x) \]

This simplifies to:

\[ \int e^x f(x) \, dx - e^x f'(x) + \int e^x f''(x) \, dx = \Psi(x) \]

### Step 4: Rearranging the equation

Now, rearranging the equation gives us:

\[ \int e^x f(x) \, dx = \Psi(x) + e^x f'(x) - \int e^x f''(x) \, dx \]

### Step 5: Solve for \(\int e^x f(x) \, dx\)

We can now isolate \(\int e^x f(x) \, dx\):

\[ \int e^x f(x) \, dx = \frac{1}{2} \left( \Psi(x) + e^x f(x) \right) \]

### Final Answer

Thus, the final result is:

\[ \int e^x f(x) \, dx = \frac{1}{2} \left( \Psi(x) + e^x f(x) \right) \]