A B737 FLT CREW EXPERIENCED AN ENG FAILURE AFTER TKOF AND LANDED UNEVENTFULLY.

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WE WERE BEGINNING OUR 3 DAY TRIP IN MDW WITH AN HR DELAY DUE TO THE REPLACEMENT OF THE #1 ENG OIL FILTER. THE ACFT HAD 2 PREVIOUS WRITE-UPS FOR THE #1 ENG OIL FILTER BYPASS LIGHT ILLUMINATING ON THE GND SO DURING OUR DELAY WE RAN THE NUMBERS FOR A SINGLE ENG FLAPS 15 DEGS RETURN TO MDW AT OUR TKOF WT (NEAR ATOG) AND REVIEWED THE PROCS AND THE ONBOARD PERFORMANCE COMPUTER. AFTER ENG START; WE DID A SHORT RUN UP OF THE #1 ENG BEFORE TAXIING AND ON THE RWY BEFORE DEP. WE DID A BLEEDS OFF TKOF AND SHORTLY AFTER RECONFIGURING THE BLEEDS; THE #1 ENG BECAME ERRATIC ON ALL PARAMETERS(N1; EGT; N2; FUEL FLOW). WHILE THE FO WAS GETTING OUT THE QRH; THE #1 ENG FAIL LIGHT BEGAN TO FLASH; THEN TURNED STEADY. THE FO RAN THE ENG FAILURE/SHUTDOWN CHKLIST IN ITS ENTIRETY. WE GOT VECTORS BACK FOR THE ILS RWY 4 AT MDW WHICH GAVE US THE BEST PERFORMANCE. I BRIEFED THE FLT ATTENDANTS AND PAX. WE DECLARED AN EMER AND HAD THE FIRE TRUCKS STANDING BY. WE LANDED FLAPS 15 DEGS SMOOTHLY ON THE TOUCHDOWN ZONE. ONCE CLR OF THE RWY; WE HAD THE FIRE TRUCKS INSPECT THE ENG AND BRAKES FOR ANY UNSAFE INDICATIONS. DURING THIS TIME; THE FO RAN THE BRAKE COOLING INFO IN THE ONBOARD PERFORMANCE COMPUTER. WHEN DETERMINED SAFE; WE TAXIED TO THE GATE WHERE THERE WERE 4 CUSTOMER SVC PEOPLE WAITING WHO HANDLED THE CUSTOMERS EXCELLENTLY. THE MECHS SAID THAT THE GEAR BOX HAD FAILED AND METAL PIECES HAD CONTAMINATED THE OIL (AFTER INSPECTING THE ENG). I SPOKE WITH THE POLICE; DISPATCH; CHIEF PLTS AND SCHEDULING BEFORE WE GOT AN ACFT SWAP AND RE-PLANED 134 PAX OF THE PREVIOUS 137 PAX AND DEPARTED AGAIN.