Math Tutor USA – Algebra – Geometry

Integration Made Easy
By Masood Amir
Calculus is a tool used to measure change or variation of a function with respect to the independent variable.
Differential Calculus:
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”.
Integral Calculus:
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 time.
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.
Techniques of Integration:
As we 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
Ist Formula of Integration (1st Rule of Integration)
Indefinite Integration:
Ist Formula of Integration (Ist Rule of Integration):
Let ? dy = ? xndx
y = xn+1/n+1 + C
Why “C”:
In the process of differentiation, we eliminate constant, as the derivative of a constant is “zero”.
So, In 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).
Example: Solve ?x3dx
Solution: x3+1/3+1 + C = x4/4 + C
Example: Solve ?(x3 + x2 + 5x + 6)dx
Solution: ?x3dx + ?x2dx + ?5xdx + ?6dx
X4/4 + x3/3 + 5×2/2+ 6x + C
Worksheet # 1
Find the Integral of the following:
?(x3-4×2+5x-6)dx ?(3×5-4×3+3×2)dx
?(ax5-bx4)dx ?(x3/2-5×4/3+3×2)dx
?(4v3x2 -2x)dx ?(3x + 5×2 –x3/2-0.4×4)dx
?(x(8x-1/2)dx ?(2-x)(4+3x)dx
?(x-3 + x-4)dx ?(2×3-3)3x4dx
?(4×7 + 3×12 -5×8 + 2x -1)dx ?(ax3 – bx2 + cx –d)dx
?(1/x3 + 2/x2 -6)dx ?(-3x-8 + 2vx)dx
?((x3 -5)(2x + 5)dx ?(7x-6 + 5vx)dx