Find `int(1)/(x+1)dx`

Options

A. `ln(x+1)+C`
B. `(x+1)^(-2)+C`
C. `(1)/((x-1)^(2))+C`
D. `ln (x-1)+C`

Answer

To solve the integral \( \int \frac{1}{x+1} \, dx \), we can follow these steps:

### Step 1: Identify the integral
We need to integrate the function \( \frac{1}{x+1} \).

### Step 2: Use substitution
Let's use a substitution to simplify the integral. We can let:
\[ t = x + 1 \]
Then, we differentiate \( t \) with respect to \( x \):
\[ dt = dx \]
This means that \( dx \) can be replaced with \( dt \).

### Step 3: Rewrite the integral
Now, we can rewrite the integral in terms of \( t \):
\[ \int \frac{1}{x+1} \, dx = \int \frac{1}{t} \, dt \]

### Step 4: Integrate
We know that the integral of \( \frac{1}{t} \) is:
\[ \int \frac{1}{t} \, dt = \ln |t| + C \]
where \( C \) is the constant of integration.

### Step 5: Substitute back
Now we substitute back \( t = x + 1 \):
\[ \ln |t| + C = \ln |x + 1| + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{1}{x+1} \, dx = \ln |x + 1| + C \]