Integration

We are given the integral: We know that: Now, evaluate : Since , this is an odd function. The integral of an odd function over a symmetric interval is always zero: Thus, the correct answer is .

Step 1: Understanding the Concept:
The problem involves an integral with an exponential term containing a logarithm in its exponent, multiplied by another exponential term. The key is to simplify the integrand using logarithmic and exponential properties before attempting integration.

Step 2: Key Formula or Approach:
Use the property of logarithms that and the fundamental property relating exponentials and natural logarithms: . In higher-level mathematics, ' ' without a specified base usually implies the natural logarithm ' '.

Step 3: Detailed Explanation:
The given integral is: Assuming denotes the natural logarithm : First, simplify the term . Using the power rule for logarithms ( ): Using the inverse property : Now, substitute this simplified expression back into the integrand: Use exponent rules to combine terms with the same base ( ): Since any non-zero number to the power of 0 is 1: The integral of a constant 1 with respect to is : (Note: The handwritten tick mark on option 4 in the image is incorrect. The mathematically sound result based on the printed text is .)

Step 4: Final Answer:
The integral evaluates to .