Question

Evaluate the integral: \[ \int_{-5}^{5} \frac{e^x + e^{-x}}{e^x - e^{-x}} \, dx. \]

• A. 0
• B. 1
• C. \( 3e^5 \)
• D. \( 2e^5 \)

Hint:

The integral of an odd function over a symmetric interval \( [-a, a] \) is always 0.

Answer 1

The Correct Option is A

Step 1: Simplify the integrand.
We start by simplifying the integrand: \[ \frac{e^x + e^{-x}}{e^x - e^{-x}}. \] This is a standard form that can be simplified as: \[ \frac{e^x + e^{-x}}{e^x - e^{-x}} = \coth(x). \]
Step 2: Recognize the symmetry.
The integrand \( \coth(x) \) is an odd function, because \( \coth(-x) = -\coth(x) \). The integral of an odd function over a symmetric interval \( [-a, a] \) is 0. Thus, the value of the integral is: \[ \int_{-5}^{5} \coth(x) \, dx = 0. \]
Step 3: Conclusion.
Thus, the value of the integral is 0, which corresponds to option (A).