 Hello friends, let's discuss the following question. It says integrate the following function. The given function is e to the power 2x minus 1 upon e to the power 2x plus 1. Let us now start the solution. Let i be the integral e to the power 2x minus 1 upon e to the power 2x plus 1 dx. Again this is equal to integral taking e to the power x common from the numerator. The numerator becomes e to the power x into e to the power x minus e to the power x minus x upon. Similarly, taking e to the power x common from the denominator, the denominator becomes e to the power x into e to the power x plus e to the power minus x dx. Now, e to the power x gets cancelled with e to the power x. Now, we see that the derivative of e to the power x plus e to the power minus x is e to the power x minus e to the power minus x. So, put e equal to e to the power x plus e to the power minus x. So, dt by dx is equal to e to the power x minus e to the power minus x. And this implies dt is equal to e to the power x minus e to the power minus x dx. Now, e to the power x minus e to the power minus x into dx is dt and t equal to e to the power x plus e to the power minus x. So, substituting all these values in the integral, the integral becomes 1 upon t dt. Now, the integral of 1 upon t dt is log t plus c. So, it is log t plus c. Now, substitute the value of t here. So, this becomes log e to the power x plus e to the power minus x plus c. Hence, the derivative of the given function is log e to the power x plus e to the power minus x plus c. And this completes the question. Bye for now. Take care. Have a good day.