If `int[(1+x)e^x f(x) +xe^x f'(x)] dx =e^x` then f (x) =

Options

A. 1
B. x
C. `1//x`
D. `e^x`

Answer

To solve the equation

\[ \int \left[(1+x)e^x f(x) + xe^x f'(x)\right] dx = e^x, \]

we will use integration by parts and properties of integrals. Let's break this down step by step.

### Step 1: Rewrite the Integral
We can separate the integral into two parts:

\[ \int (1+x)e^x f(x) \, dx + \int xe^x f'(x) \, dx = e^x. \]

### Step 2: Apply Integration by Parts
For the second integral \(\int xe^x f'(x) \, dx\), we will apply integration by parts. Let:

- \(u = x\) (which means \(du = dx\))
- \(dv = e^x f'(x) dx\) (which means \(v = e^x f(x)\))

Using integration by parts, we have:

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

Thus,

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

### Step 3: Substitute Back
Now, substituting this back into the original equation, we have:

\[ \int (1+x)e^x f(x) \, dx + \left[x e^x f(x) - \int e^x f(x) \, dx\right] = e^x. \]

### Step 4: Combine the Integrals
Now, combine the integrals:

\[ \int (1+x)e^x f(x) \, dx + x e^x f(x) - \int e^x f(x) \, dx = e^x. \]

### Step 5: Simplify the Equation
Notice that the integral terms can be simplified. The integral \(\int e^x f(x) \, dx\) appears on both sides of the equation. Thus, we can cancel them out:

\[ \int (1+x)e^x f(x) \, dx + x e^x f(x) - \int e^x f(x) \, dx = e^x. \]

This simplifies to:

\[ \int (1+x)e^x f(x) \, dx = e^x - x e^x f(x). \]

### Step 6: Solve for \(f(x)\)
Now, we can isolate \(f(x)\):

\[ (1+x)e^x f(x) = e^x. \]

Dividing both sides by \(e^x (1+x)\):

\[ f(x) = \frac{e^x}{(1+x)e^x} = \frac{1}{1+x}. \]

Thus, the solution for \(f(x)\) is:

\[ f(x) = \frac{1}{1+x}. \]

### Final Answer
\[ f(x) = \frac{1}{1+x}. \]