Calculus is a tool used to measure change or
variation of a function with respect to the independent variable.
It is the branch of calculus used to measure change
or variation of a function in a very small interval of time, the techniques use
to measure such changes is called “differentiation”.
It is the branch of calculus used to measure changes
or variation over an interval of independent variable, e.g to find length of
curve, the area of region and the volume of a solid in a specified period of
The technique used to measure such changes or
variation is called “Integration”
or “Antiderivatives”. It a
reverse process of differentiation.
Mathematically, Integration is defined as “ If f’(x)
represents the differential coefficient of f(x), then the problem of
integration is given f’(x), find f(x) or given dy/dx, find y.
Notation: ”∫” is used to show the
integration, it is a symbol of “S” derived from the word “Sum”. i.e.
Integration is a process in which we have to sum up the derivatives over a
specified interval and to find the function.
know that integration is the reverse process of differentiation, our problem is
to find the function f(x) or Y, when f’(X) or dy/dx is given.
dy/dx = f’(X)
∫dy = ∫f ’(X)dx
Y= f(x) is our solution
Formula of Integration (1st Rule of Integration)
Formula of Integration (Ist Rule of Integration):
Let ∫ dy = ∫ xndx
y = xn+1/n+1
process of differentiation, we eliminate constant, as the derivative of a
constant is “zero”.
functions like Xn, Xn+ 6, Xn + 3 , Xn – K, the derivatives of all of them is
Xn-1, in finding the anti derivative of Xn-1, we put a constant “C”, as we don’t
know which constant was present in the original function, and this can be found
If we have initial boundary values (Definite Integral).