integrate dx over e to the power of x minus e tothe power of negative x

Question

Find the following indefinite integrals

Answer 1

Integral dx / e^x - e^-x

multiply both numerator and denominator by e^x

or express e^-x = 1/e^x and take LCM

= Integral e^x dx / e^2x - 1

put e^x = t

e^x dx = dt

Integral dt / t^2 - 1

1/2 log [(t-1)/(t+1)] + c

1/2 log [(e^x-1)/(e^x+1)] +c