 Ch 6. Integrals Multimedia Engineering Math IndefiniteIntegral Area DefiniteIntegral FundamentalTheorem SubstitutionRule
 Chapter 1. Limits 2. Derivatives I 3. Derivatives II 4. Mean Value 5. Curve Sketching 6. Integrals 7. Inverse Functions 8. Integration Tech. 9. Integrate App. 10. Parametric Eqs. 11. Polar Coord. 12. Series Appendix Basic Math Units Search eBooks Dynamics Statics Mechanics Fluids Thermodynamics Math Author(s): Hengzhong Wen Chean Chin Ngo Meirong Huang Kurt Gramoll ©Kurt Gramoll MATHEMATICS - THEORY

This section discusses the concept of antiderivative and the theorms related to it.

Antiderivative

A derivative provides an easy way to find the rate of changes for a function. On the other hand, an antiderivative is a way to find the function of a given derivative.

An antiderivative of a function f is any function F such that F' = f. It is a reverse process of differentiation.

Indefinite Integral

The notation used to refer to antiderivative is the indefinite integral. ∫f(x)dx means the antiderivative of f with respect to x . If F(x) is an antiderivative of f(x), then

∫f(x)dx = F + c

in which c is an arbitrary constant.

 Function Antiderivative ∫cf(x)dx cF(x) + c ∫xndx   (n ≠ -1) xn+1/(n + 1) + c ∫(f(x) + g(x))dx F(x) + G(x) + c ∫cosxdx sinx + c ∫sinxdx -cosx + c ∫sec2xdx tanx + c ∫secxtanxdx secx + c ∫cscxcotxdx -cscx + c ∫csc2xdx -cotx + c

The table on the left gives various examples of indefinite integrals.

The method to calculate the antiderivative or indefinite integral can be better understand by finding the antiderivative of function f '(x) = x2 when f(1) = 7/3.

The indefinite integral rules gives that The antiderivative of the given function f '(x) = x2 can be expressed as In order to determine the constant c, substituting f(1) = 7/3 into the above expression.

f(1) = 1/3 + c = 7/3 Family of Curves for f(x) = x3/3 + c

Rearranging the above equation gives

c = 2

Therefore, the solution of this example is

f(x) = x3/3 + 2

Notice that the f(x) = x3/3 + c is a family of functions and a specific function depends of the given intial condition or boundary condition. Its graphic is shown on the left.