Question

The value of \[ \int_0^1 \frac{dx}{e^x + e^{-x}} \] is :

• A. $-\frac{\pi}{4}$
• B. $\frac{\pi}{4}$
• C. $\tan^{-1} e - \frac{\pi}{4}$
• D. $\tan^{-1} e$

Hint:

For integrals involving hyperbolic functions, use known identities and simplify to standard integral forms for faster solutions.

Answer 1

The Correct Option is B

The given integral is simplified using a substitution method. The integral is defined as: \[ I = \int_0^1 \frac{dx}{e^x + e^{-x}} \] By utilizing the identity $e^x + e^{-x} = 2 \cosh x$, the integral transforms to: \[ I = \int_0^1 \frac{dx}{2 \cosh x} \] This represents a standard integral with the solution: \[ I = \frac{\pi}{4} \] Therefore, the correct answer is $\frac{\pi}{4}$.