Integral with variable upper limit of integration Let f ( x) be a continuous function, given in a segment [ a , b ], then for any x [ a , b ] the function exists. This function is given as an integral with variable upper limit of integration in the right-hand part of the equality. All rules and properties of a definite integral apply to an integral with … [Read more...]

## Some definite integrals

Some definite integrals Back … [Read more...]

## Geometrical and mechanical applications of definite integral

Geometrical and mechanical applications of definite integral Volume of revolution body. Work of variable force. A definite integral has numerous applications in mathematics, mechanics, physics, astronomy, engineering and other fields of human activities. We’ll consider here only two examples, illustrating possibilities of this apparatus. A volume of revolution … [Read more...]

## Basic properties of definite integral

Basic properties of definite integral Back … [Read more...]

## Definite integral. Newton – Leibniz formula

Definite integral. Newton – Leibniz formula Curvilinear trapezoid. Area of a curvilinear trapezoid. Definite integral. Limits of integration. Integrand. Newton-Leibniz formula. Consider a continuous function y = f ( x ), given on a segment [a, b] and saving its sign on this segment ( Fig.8 ). The figure, bounded by a graph of this function, a segment [a, b] and … [Read more...]

## Some indefinite integrals of elementary functions

Some indefinite integrals of elementary functions In the last integral the integration segment doesn’t contain x = 0. Below we’ll ignore a constant of integration C. Back … [Read more...]

## Integration methods

Integration methods Integration by parts. Integration by substitution ( exchange ). Integration by parts. If functions u ( x ) and v ( x ) have continuous first derivatives and the integral v ( x ) du ( x ) exists, then the integral u ( x ) dv ( x ) also exists and the equality u ( x ) dv ( x ) = u ( x ) • v ( x ) – v ( x ) du ( x ) takes place, or … [Read more...]

## Basic properties of indefinite integral

Basic properties of indefinite integral If a function f ( x ) has a primitive on a segment X , and k – a number, then If functions f ( x ) and g ( x ) have primitives on a segment X , then If a function f ( x ) has a primitive on a segment X , then for interior points of this segment: If f ( x ) is a continuous function on a … [Read more...]

## Primitive. Indefinite integral

Primitive. Indefinite integral Primitive. Finding of primitive: infinite set of solutions. Indefinite integral. Constant of integration. Primitive. A continuous function F ( x ) is called a primitive for a function f ( x ) on a segment X , if for each F’ ( x ) = f ( x ). E x a m p l e . The function F ( x ) = x 3 is a primitive for the function f ( x … [Read more...]

## Convexity, concavity and inflexion points of a function

Convexity, concavity and inflexion points of a function The second derivative. Convex and concave function. Sufficient condition of concavity ( convexity ) of a function. Inflexion point. The second derivative. If a derivative f ' ( x ) of a function f ( x ) is differentiable in the point ( x0 ), then its derivative is called the second derivative of the function f ( x ) … [Read more...]