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Does sound like a smoked relay, switch, or integration relay issue. If you just bought the car, dealership should be eating the diagnostic and repair charges as a sign of future business. If not, take the car elsewhere - as the repair estimate for the integration relay is "reasonable", but the gear box lamp is not, assuming this is talking about the light for the gear position indicator plate.
For that - you can DIY if you feel comfortable pulling trim panels off the car - basically work from the top down, pulling trim off until you expose the wire leading to the bulb under the indicator panel. I don't remember if it is a filament bulb or LED, but it is a little lamp in there.
Center finish panel subassembly (around radio), instrument cluster center console finish panel (around shifter), instrument panel lower box finish panel (around shifter base), this will expose shifter assembly, look on the left side of the shifter, right below the indicator strip - should see the wire, bulb holder, and tiny lamp that needs to be changed.
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2002 Corolla S, 1.8L 1ZZ-FE VVT-i
2003 Matrix XRS, 1.8L 2ZZ-GE, VVTL-i (RIP)
2009 Matrix XRS, 2.4L 2AZ-FE VVT-i
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