Integral calculus and a core concept in mathematics analysis. Usually divided into The definite integral And The indefinite integral Two.Intuitively, for a given positive real valued functionF (x)In a real number, interval [a, b] of the definite integral:
Can be understood as in Oxy coordinate plane, a curve (x, f (x)), x=a, x=b and X axis line enclosed area of trapezoid with curved edge value (a certain real valued).
The integral formula of definite integral mainly include the following: integral, containing ax+b (a+bx) with √ integral, containing x^2 + α ^2 containing ax^2+b (a>0) integral, integral, containing V (a^2+x^2) (a>0) integral, containing V (a^2-x^2) (a>0) integral, containing V (|a|x^2+bx+c) (a = 0), containing the integral trigonometric function, inverse trigonometric function integral containing integrals containing exponential function, logarithmic function integral, integral, containing containing hyperbolic function integral.
Integral formula with ax+b mainly in the following categories:
Containing or (a+bx) integral formula only contains the following categories:
The integrand contains V (a^2+x^2) (a>0) of the integral:
The integrand contains V (a^2-x^2) (a>0) of the integral:
Containing or (|a|x^2+bx+c) (a = 0) integral
The integrand contains V (|a|x^2+bx+c) (a = 0) with the integral[2-4]
By the integral formula with trigonometric function in the:
By the integral formula of function contains anti trigonometric functions are:
The integrand contains a formula of exponential function:
The integrand contains a formula of logarithmic function:
The integrand contains a hyperbolic function of integral formula:
There are several integral formula
Usually the integral sense can satisfy some basic properties. The integral region
In the Riemann integral sense represents an interval, in the sense of Lebesgue integral representation of a measurable set. Integral nature: linear, the number of maxima, minima, absolute continuity of the integral absolute value, etc..
The integral is linear. If a functionFIntegrable, then it is multiplied by a constant still integrable. If the functionFAndGIntegrable, then their sum and difference also integrable.
If a function f in a range of Riemann integrable, and on this interval greater than or equal to zero. So it is in the interval of integration is greater than or equal to zero. If f Lebesgue integrable and almost always greater than or equal to zero, then the Lebesgue integral which is greater than or equal to zero. As a corollary, if two
The integrable functions f and G compared, f (almost) is always less than or equal to g, then f (Lebesgue) integral is equal to or less than g (Lebesgue) integral.[5-6]
If the Riemann integrable non negative function in F
The integral is equal to 0, then in addition to a finite number of points, f = 0. If Lebesgue integrable non negative function in F
The integral is equal to 0, then the f almost everywhere for 0. If
Measure of elements in Aμ (A)The integral is equal to 0, then any integrable function on A is equal to 0.[5-6]
Integral representation of functions the overall nature function in a region of the value function, change a point not to change its integral value. The function of Riemann integratiable, change the value of limited point, the integral invariant. The function is Lebesgue integrable, a measure for the function of the 0 set of values change, does not affect its integral value. If the two function almost everywhere the same, then their same points. If the
Any element in the A, integrable function f in A on the integral equals (greater than or equal to) integrable function g in A integral, then f almost everywhere equal to (greater than or equal to G).
Integral types and the following categories:
- Riemann integral
- Darboux integral
- Lebesgue integral
- Riemann Steele Jisi integral
- Numerical integration
The first step, open the Word2010 document window, switch to "insert" Function area . In the "symbol" group click "formula" button (non "formula" drop-down triangle button).
The second step, create a blank formula in the framework of the Word2010 document, in the "tools / design formula of" functional area, click the "structure" in a group of "integral" button. Select the integral form suitable in the integral structure in the list of open.
Third step, into the integral structure in the blank formula in integral structure frame, click the placeholder box and enter the specific value or formula symbols can be.