Integral Calculus

CALCULUS Note: This compilation of the definitions and examples from Chapters 1-3 of Integral Calculus which are shown below were adapted from mathalino.com. No intention to violate copyright policies. Chapter 1. Fundamental theorems of Calculus 

Indefinite Integrals

If F(x) is a function whose derivative F'(x) = f(x) on certain interval of the x -axis, then F(x) is called the anti-derivative of indefinite integral f(x). When we integrate the d ifferential of a function we get that function plus an arbitrary constant. In symbols we write

 () ()   ∫

where the symbol , called the integral sign, specifies the operation of integration upon f(x) dx; that is, we are to find a function whose derivative is f(x) or whose differential is f(x) dx. The dx tells us that the variable of integration is x.

Integration Formulas In these formulas, u and v denote differentiable functions of some independent variable (say (sa y x) and a, n, and C are constants. 1. The integral of the differential of of a function u is u plus an arbitrary arbitrary constant C (the definition of of an integral).

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

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

4. If n is not equal to minus one, the integral 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. Formula .

Definite Integral

Chapter 2. Fundamental Integration Formula

Logarithmic Functions | Fundamental Integration Formulas

Trigonometric Functions | Fundamental Integration Formulas

Inverse Trigonometric Functions | Fundamental Integration Formulas

Chapter 3. Techniques in Integration    

Integration by Parts Integration by Substitution Integration of Rational Fractions Change of Limits with Change of Variable

Integration by Parts

Integration by Substitution There are two types of substitution: algebraic substitution and trigonometric substitution. 

In algebraic substitution we replace the variable of integration by a function of a new ne w variable. A change in the variable on integration often reduces an integrand to an easier integrable form.

Trigonometric Substitution | Techniques of Integration

Integration of Rational Fractions | Techniques of Integration

Chapter 4 - Applications of Integration 

Plane Areas in Rectangular Coordinates

Example 2 | Plane Areas in Rectangular Coordinates

Example 3 | Plane Areas in Rectangular Coordinates

Example 4 | Plane Areas in Rectangular Coordinates

Example 5 | Plane Areas in Rectangular Coordinates

Example 6 | Plane Areas in Rectangular Coordinates