__Integration Made Easy__

__By Masood Amir__

__Calculus:__

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 = ∫ x^{n}dx

**y = x**^{n+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 X^{n}, X^{n}+ 6, X^{n} + 3 , X^{n} – K, the derivatives of all of them is
X^{n-1}, in finding the anti derivative of X^{n-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 ∫x^{3}dx

Solution:
**x**^{3+1}/3+1 + C = x^{4}**/**4
+ C

Example:
Solve ∫(x^{3 }+ x^{2} + 5x + 6)dx

Solution:** ** ∫x^{3}dx
+ ∫x^{2}dx + ∫5xdx + ∫6dx

**
X**^{4}/4 + x^{3}/3 + 5x^{2}/2+ 6x + C

__Worksheet
# 1__

__ __

### Find the Integral of the following:

__SET 1 __

∫(x^{3}-4x^{2}+5x-6)dx **∫(3x**^{5}-4x^{3}+3x^{2})dx

∫(ax^{5}-bx^{4})dx **∫(**^{x3/2}-5x^{4/3}+3x^{2})dx

∫(^{4}√3x^{2} -2x)dx **∫(3x + 5x**^{2}
–x^{3}/2-0.4x^{4})dx

∫(x(8x-**1/2**)dx **∫(2-x)(4+3x)dx**

∫(x^{-3} +
x^{-4})dx **∫(2x**^{3}-3)3x^{4}dx

∫(4x^{7} +
3x^{12} -5x^{8} + 2x -1)dx **∫(ax**^{3} –
bx^{2} + cx –d)dx

∫(1/x^{3}
+ 2/x^{2} -6)dx **∫(-3x**^{-8}
+ 2√x)dx

∫((x^{3}
-5)(2x + 5)dx **∫(7x**^{-6}
+ 5√x)dx