What is `int ln (x^(2)) dx` equal to ?

Options

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

Explanation

To solve the integral \( \int \ln(x^2) \, dx \), we can follow these steps:

### Step 1: Simplify the integrand
We start by using the property of logarithms that states \( \ln(a^b) = b \ln(a) \). Thus, we can rewrite \( \ln(x^2) \) as:
\[ \ln(x^2) = 2 \ln(x) \]
So, the integral becomes:
\[ \int \ln(x^2) \, dx = \int 2 \ln(x) \, dx \]

### Step 2: Apply integration by parts
To integrate \( 2 \ln(x) \), we will use integration by parts. The formula for integration by parts is:
\[ \int u \, dv = uv - \int v \, du \]
We can choose:
- \( u = \ln(x) \) → \( du = \frac{1}{x} \, dx \)
- \( dv = 2 \, dx \) → \( v = 2x \)

### Step 3: Substitute into the integration by parts formula
Now, we substitute \( u \), \( du \), \( v \), and \( dv \) into the integration by parts formula:
\[ \int 2 \ln(x) \, dx = 2x \ln(x) - \int 2x \cdot \frac{1}{x} \, dx \]
This simplifies to:
\[ \int 2 \ln(x) \, dx = 2x \ln(x) - \int 2 \, dx \]

### Step 4: Integrate the remaining integral
Now, we compute the remaining integral:
\[ \int 2 \, dx = 2x \]
So we have:
\[ \int 2 \ln(x) \, dx = 2x \ln(x) - 2x \]

### Step 5: Add the constant of integration
Finally, we add the constant of integration \( C \):
\[ \int \ln(x^2) \, dx = 2x \ln(x) - 2x + C \]

### Final Answer
Thus, the result of the integral is:
\[ \int \ln(x^2) \, dx = 2x \ln(x) - 2x + C \]

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