# 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.

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

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

- 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).

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

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

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

- If
*n*is not equal to minus one, the integral of*u*is obtained by adding one to the exponent and divided by the new exponent. This is called the^{n}du**General Power Formula**.

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