The brief
The X1 had stopped starting reliably, sometimes nothing, sometimes a whirr or a grind instead of the engine turning over, even with a healthy battery. He brought it in. When the battery's good and you're getting that whirr or grind, the starter motor is the prime suspect. The starter motor is the electric motor that spins the engine over fast enough to fire. It has brushes, a solenoid that throws the drive gear into the flywheel, and bearings, all of which wear. When it goes, the solenoid sticks or the drive doesn't engage, that's the click or the whirr with no crank, or the gear grinds against the flywheel instead of meshing cleanly, that's the grinding. A worn starter doesn't recover, and an intermittent one will leave you stranded, so it needs replacing.
The diagnosis
A check confirmed it, the battery and the charging were fine, but the starter was drawing badly and not engaging cleanly, the solenoid and the drive gear worn, exactly what makes that whirr and grind. The flywheel ring gear was inspected and was fine. That's a starter replacement, you don't rebuild one on the car, so the call was a complete starter motor, fitted and tested.
The work
The bits in the way came off, the old starter motor unbolted and removed, and a new genuine BMW-spec starter fitted, the wiring and the earth checked and reconnected cleanly, every fastener torqued to spec. The engine was cranked and started to confirm the new starter spun it over strongly and the drive engaged and released cleanly. A road test confirmed a strong, clean crank every time, no whirr, no grind.
The outcome
Strong, clean cranking, the engine firing first time, no whirr, no grind, and no more being stuck. The X1 went home starting properly again. A worn starter only gets more unreliable until it strands you, so changing it put the starting back where it should be, the job you want done before the morning it won't go at all.