How long does semen really live?

My SO and I are TTC and I ovulated on the 10th. However due to our busy conflicting schedules we were only able to have sex on Aug 4th! 😒😒😒😒😒😒😒😒. That would make it 6 days of the semen being inside me. I did have some

Brown spotting on Monday of this week or at least that's what I thought it was. Can anybody help?!