Question

\( \int \log x^2 \, dx = \)

• A. \( \log x^2 + x + c \)
• B. \( x\log x^2 - 1 + c \)
• C. \( x\log x^2 + x + c \)
• D. \( x\log x^2 - 2x + c \)

Hint:

Always simplify logarithmic expressions first using identities before integration.

Answer 1

The Correct Option is D

Step 1: Use logarithmic identity.
\[ \log x^2 = 2\log x. \]
So, the integral becomes:
\[ \int \log x^2 \, dx = \int 2\log x \, dx. \]

Step 2: Take constant outside.
\[ = 2 \int \log x \, dx. \]

Step 3: Use standard integral formula.
\[ \int \log x \, dx = x\log x - x + C. \]

Step 4: Substitute in expression.
\[ 2(x\log x - x). \]

Step 5: Simplify.
\[ = 2x\log x - 2x. \]

Step 6: Express in terms of \( \log x^2 \).
Since \( 2\log x = \log x^2 \), we get:
\[ x\log x^2 - 2x. \]

Step 7: Final conclusion.
\[ \boxed{x\log x^2 - 2x + C}. \]