# 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):

**If y = x ^{n}**

**∫ 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

**+ 6, X**

^{n}**+ 3 , X**

^{n}**– K, the derivatives of all of them is X**

^{n}**, 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).**

^{n-1},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 **

a. **∫(x ^{3}-4x^{2}+5x-6)dx **

**∫(3x**^{5}-4x^{3}+3x^{2})dx

c. **∫(ax ^{5}-bx^{4})dx **

**∫(**^{x3/2}-5x^{4/3}+3x^{2})dx

e.**∫( ^{4}√3x^{2} -2x)dx **

**∫(3x + 5x**^{2}–x^{3}/2-0.4x^{4})dx

g. **∫(x(8x-****1/2****)dx **

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

k. **∫(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

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