Question

\( \int \frac{e^{2025x} + e^{-2025x}}{e^{2025x} + e^{-2025x}} dx = \)_____ + C

• A. \( e \log|e^x + e^{-x}| \)
• B. \( \frac{1}{e} \log|e^x + e^{-x}| \)
• C. \( \log|e^x + e^{-x}| \)
• D. \( -\frac{1}{e} \log|e^x + e^{-x}| \)

Hint:

If numerator resembles derivative of denominator, use substitution immediately.

Answer 1

The Correct Option is B

Concept: Use substitution: \[ u = e^x + e^{-x} \]
Step 1: Differentiate. \[ \frac{du}{dx} = e^x - e^{-x} \]
Step 2: Expression becomes: \[ \int \frac{du}{u} \]
Step 3: \[ = \log|u| + C = \log|e^x + e^{-x}| + C \] Scaling gives: \[ = \frac{1}{e} \log|e^x + e^{-x}| + C \]