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welcome to our lectern line in this video were going to show you how to take the integral of a complex function so again we start with Z being equal to a plus bi therefore e to the ZX can be written as a to the a plus bi times X so when we separate it as follows a to the ax times e or e to the ax times e to the X DX now we can take the integral if you use integration by parts so we can say that the integral of U DV is equal to u times V minus the integral of VDU we just have to figure out what were going to call you and what were going to call DV we probably want to call E - the i VX u so let u equal e to the I B X that means that D U is equal to C here e to the I be x times I times B times DX which means that DV can be written as e to the ax DX and therefore V is going to be equal to 1 over a e to the ax so now were ready to try and integrate this so lets plug in what we have the integral of e to the ax e to the I B X DX is equal to u times V and U is e to the IPX and V is 1 ove