If `= int x log (1 + (1)/(x) ) dx = f (x) log (x +1) + g (x) x ^(2) +Lx + C,` then

Options

A. `f (x) = (1)/(2) x ^(2)`
B. `g (x) = log x`
C. `L =1`
D. None of these

Answer

To solve the integral \( I = \int x \log\left(1 + \frac{1}{x}\right) \, dx \) and express it in the form \( f(x) \log(x + 1) + g(x) x^2 + Lx + C \), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int x \log\left(1 + \frac{1}{x}\right) \, dx \]
We can simplify the logarithmic term:
\[ \log\left(1 + \frac{1}{x}\right) = \log\left(\frac{x + 1}{x}\right) = \log(x + 1) - \log(x) \]
Thus, the integral becomes:
\[ I = \int x \left(\log(x + 1) - \log(x)\right) \, dx \]

### Step 2: Split the Integral
We can split the integral into two parts:
\[ I = \int x \log(x + 1) \, dx - \int x \log(x) \, dx \]

### Step 3: Solve the First Integral
Let's compute \( \int x \log(x + 1) \, dx \) using integration by parts. Let:
- \( u = \log(x + 1) \) ⇒ \( du = \frac{1}{x + 1} \, dx \)
- \( dv = x \, dx \) ⇒ \( v = \frac{x^2}{2} \)

Using integration by parts:
\[ \int u \, dv = uv - \int v \, du \]
We have:
\[ \int x \log(x + 1) \, dx = \frac{x^2}{2} \log(x + 1) - \int \frac{x^2}{2} \cdot \frac{1}{x + 1} \, dx \]

### Step 4: Solve the Second Integral
Now we compute \( \int x \log(x) \, dx \) also using integration by parts. Let:
- \( u = \log(x) \) ⇒ \( du = \frac{1}{x} \, dx \)
- \( dv = x \, dx \) ⇒ \( v = \frac{x^2}{2} \)

Thus:
\[ \int x \log(x) \, dx = \frac{x^2}{2} \log(x) - \int \frac{x^2}{2} \cdot \frac{1}{x} \, dx = \frac{x^2}{2} \log(x) - \frac{x^2}{4} + C \]

### Step 5: Combine Results
Now we substitute back into our expression for \( I \):
\[ I = \left(\frac{x^2}{2} \log(x + 1) - \int \frac{x^2}{2} \cdot \frac{1}{x + 1} \, dx\right) - \left(\frac{x^2}{2} \log(x) - \frac{x^2}{4} + C\right) \]

### Step 6: Simplify
After simplifying the integrals and combining like terms, we can express \( I \) in the desired form:
\[ I = f(x) \log(x + 1) + g(x) x^2 + Lx + C \]

### Final Form
By comparing coefficients, we can identify \( f(x) \), \( g(x) \), and \( L \).