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Kohli captained RCB in 140 matches for eight seasons since 2013.
Kohli was annoyed with Maxwell for mocking him following his shoulder injury during the 2017 Border-Gavaskar Trophy.
The India wicketkeeper-batter announced his decision with a post on X.
Kohli also hailed Karthik as a technically 'correct' and versatile batter.
The loss ended RCB's six-match winning streak as their search for a maiden IPL title continues.
RR restricted RCB to 172/8, a total they overhauled with six balls to spare in the Eliminator at the Narendra Modi Stadium.
Swapnil credited RCB head coach Andy Flower for keeping the faith in him in the lead-up to the auction.
Dhoni was dismissed off the very next delivery after the ball was replaced.
CSK needed 17 off the final over to qualify, but Dayal held his nerve despite being hit for a towering six by Dhoni off the first ball to seal it in the end for RCB.
RCB's sixth win on the trot also meant it was the end of the road for five-time champions CSK.
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