Properties of Integrals

Integration Formulas

In these formulas, u and v denote differentiable functions of some independent variable (say x) and a, n, and C are constants.

  1. The integral of the differential of a function u is u plus an arbitrary constant C (the definition of an integral).

    $$\displaystyle \int du = u + C$$

  2. The integral of a constant times the differential of the function. (A constant may be written before the integral sign but not a variable factor).

    $$\displaystyle \int a \, du = a\int du$$

  3. The integral of the sum of a finite number of differentials is the sum of their integrals.

    $$\displaystyle \int (du + dv + ... + dz) = \int du + \int dv + ... + \int dz$$

  4. If n is not equal to minus one, the integral of un du is obtained by adding one to the exponent and divided by the new exponent. This is called the General Power Formula.

    $$\displaystyle \int u^n \, du = \dfrac{u^{n + 1}}{n + 1} + C; \, n \neq -1$$