FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.573.5 INTEGRATION
Definition:
For differential caculus we know. dxd(x 2x^2 )= Here for a certain given function x^2 , we have calculated its
differential co-efficient w.r.t.x. The reverse process is known as integration, i.e. a certain function is given to
us and we are required to find another function of the same variable whose differential co-eficient (w. r. t.
the same variable) is the given function. For example, let the given function be 2x, we are to find another
function (of the same variable x) whose differential co-efficient: (w. r. t. x) is 2x. Now the function will be x^2
and we shall say that the integration of 2x w. r. t. x is x^2. Sign. Let f(x) and φ (x) be two functions of x so that
derivative of φ (x) w. r. t. x. is f(x), i.e.,
d (x)
dxφ =f(x)^ then the integral of f(x) is φ(x), which is expressed by attaching the sign of integration before f(x)
and attaching dx after f(x), indicating that x is the variable of integration.
So if dxdφ(x)=f(x) then f(x)dx (x).= φ
The function f(x) which is to be integrated is called the integrand.
Here f(x)dx indicates in the indefinite integral of (fx) w. r. t. x.
Note. (i) We find integration is the inverse process of differention.
(ii) dxd ( ) and ( ) dx, the symbols are reverse to each other.
Constant of Integration
We know dxdx 2x; (x 5) (x ) (5) 2x 0 2xand (x c) 2x^2 = dxd^2 + =dxd^2 +dxd = + = dxd^2 + =
In general, if dxdφ(x) f(x),then [ (x) c] f(x)= dxd φ + =
f(x) dx = φ^ (x) + c, where c is a constant. (c is also known as constant of integration)
General Theorems concerning Integration : (A) The integral of the algebraic sum of a finite
Number of functions is equal to the algebraic sum of their integrals. In
(u 1 ±^ u 2 ± u 3 ........±^ u n) dx = u 1 -dx^ ± u 2 -dx^ ±.....^ ±^ un^ dx.
Where u 1 , u 2 , un are all functions of x or constants.