Evaluate : `int((1)/(ax + b)) dx.`

Options

A. `log_(e)(ax+b)+C`
B. `a log_(e) (ax+b)+C`
C. `C+(1)/(a) log_(e)(ax+b)`
D. `(1)/(b)log_(e)(ax+b)+C`

Explanation

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

### Step 1: Substitution
Let \( u = ax + b \).

### Step 2: Differentiate \( u \)
Now, differentiate both sides with respect to \( x \):
\[ \frac{du}{dx} = a \implies du = a \, dx \implies dx = \frac{du}{a} \]

### Step 3: Rewrite the Integral
Substituting \( u \) and \( dx \) into the integral, we get:
\[ \int \frac{1}{ax + b} \, dx = \int \frac{1}{u} \cdot \frac{du}{a} \]

### Step 4: Factor out the Constant
Factor out the constant \( \frac{1}{a} \):
\[ = \frac{1}{a} \int \frac{1}{u} \, du \]

### Step 5: Integrate
Now, we can integrate:
\[ = \frac{1}{a} \ln |u| + C \]

### Step 6: Substitute Back
Substituting back \( u = ax + b \):
\[ = \frac{1}{a} \ln |ax + b| + C \]

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

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