Integrals of Polynomials

The integral of any polynomial is the sum of the integrals of its terms. A general term of a polynomial can be written 任何多项式的积分等于其各项积分之和。多项式的一般项可以表示为 and the indefinite integral of that term is 该术语的不定积分是 where a and C are constants. The expression applies for both positive and negative values of n except for the special case of n= -1. In the examples, C is set equal to zero. If definite limits are set for the integration, it is called a definite integral. 其中a和C是常数。该表达式适用于n的正负值，除了n=-1的特殊情况。在示例中，C被设为零。如果对积分设定确定的上下限，则称为定积分。

Case for n = -1 n = -1 的情况

Applications

Definite Integrals of Polynomials

The definite integral of any polynomial is the sum of the integrals of its terms. A general term of a polynomial can be written 任何多项式的定积分等于其各项积分之和。多项式的一般项可以表示为 and the definite integral of that term is 该术语的定积分是 where b and c are constants, called the limits of the integral. The procedure is basically the same as in the indefinite integral except for the evaluation at the two limits. 其中，b 和 c 是常数，称为积分的限。该过程基本上与不定积分相同，只是在两个限上进行评估。

Case for n = -1 n = -1 的情况

Applications

Integral of 1/x

As a special case of the integral of a polynomial, the integral of 1/x gives a natural logarithm.

where C is a constant of integration.

For the definite integral case:

Applications

Applications: Integral of 1/x

This integral appears in processes where the rate of change of a variable is proportional to the variable itself. This occurs in radioactive decay:

Another example is in the calculation of the work done by a gas during an isothermal process. In this case the pressure is inversely proportional to the gas volume V, leading to the work integral

Polynomial Integrals: Applications

Definite integrals of polynomials show up in many physical applications:

Potential energy of stretched spring:

Moment of Inertia of rod, cylinder, or sphere

Center of mass of rod.

Motion equations for constant acceleration.

Gravitational potential energy at large heights.

Calculating energy stored in a capacitor.

The calculation of voltage difference near a point charge involves a polynomial integral with negative power:

An example with a negative exponent is the calculation of work in an adiabatic process.