Question

`int (dx)/(e^x + e^(-x))` is equal to ______.

Options

• `tan^-1 (e^x) + c`

• `tan^-1 (e^-x) + c`

• `log (e^x - e^-x) + c`

• `log (e^x + e^-x) + c`

Answer 1

`int (dx)/(e^x + e^(-x))` is equal to `underlinebb(tan^-1 (e^x) + c)`.

Explanation:

Let I = `int (dx)/(e^x + e^-x)`

= `int (dx)/(e^x + 1/e^x)`

= `int (e^x   dx)/(e^(2x) + 1)`

Put `e^x = t, e^x  dx = dt`

I = `int 1/(t^2 + 1) dt = tan^-1 t + c = tan^-1 (e^x) + c`