1.8.2—Constant of integration

After integrating a derivative, +C represents the unknown vertical position. A condition such as y=4 when x=1 determines C.

Integrate first, then substitute the given coordinate or initial value. In a motion problem, use the condition on displacement or velocity at the stated time.

If dy/dx=6x−2 and y(1)=5, then y=3x²−2x+C, giving C=4 and y=3x²−2x+4.

Setting C=0 assumes a particular origin that the question may not give; it is not a harmless simplification.