The purpose of long division in this integration is to simplify the expression and make it easier to integrate.
To perform long division, divide the numerator (e^x) by the denominator (e^{2x} + 1) using the same steps as traditional long division. This will result in a quotient and a remainder.
The remainder can be written as a fraction with the original denominator (e^{2x} + 1). This fraction can then be integrated separately.
It is possible to integrate the original expression without performing long division, but it will be more complicated and time-consuming. Long division simplifies the integration process.
Yes, if the degree of the numerator (e^x) is less than the degree of the denominator (e^{2x} + 1), then long division is not necessary and the integration can be done directly.