Integration Rules

Integration rules is one of the most important topic within calculus. This page is based on integration rules which is closely related to the study of differentiation rules. First a brief description is given on integration and differentiation and then further integration rules and differentiation rules are provided with explained examples for your better understanding. Grab this learning here and gain quality calculus help.

I like to share this Integers Rules with you all through my article.

Calculus is a branch in mathematics focused on limits, functions, differentiation, integration, and infinite series. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. The differential calculus is a measure of how a function changes as its input changes. The process of finding a derivative is called differentiation. The reverse process of differentiation is called anti-differentiation.                                                                                                                                                                                                                    Source Wikipedia
Integration Rules - Examples

Below are the integration rules provided with explained examples. All the rules are given with a solved  example following for your better understanding:

Integration rule 1:

Constant rule: `int` ` [b f(x)]` dx  = `b ` `int` ` [f(x)] dx`

Example:           If f(x)  = 6x.  The integration of f(x) = int f(x) = 6 `int` (x dx) = `6 (x^2/2) ` = 3x2

Integration rule 2:

Integration sum rule :  `int [f(x) + g(x)] dx = int f(x) dx + int g(x)dx`

Example:                     f(x) = x2 + 5

The integration of f(x) = `int ` f(x) dx = ` x^3/3` + 5x + c

Integration rule 3:

Integration Difference rule:   `int [f(x) - g(x)]dx = int f(x) dx - int g(x) dx`

Example:                     f(x) = x3 - 5x

The integration of f(x) =`int` f (x) dx = `x^4/4 - 5(x^2/2) ` + c

Integration rule 4:

Integration power rule:  `int` (x)n dx =   ` (x^(n+1)/(n+1) + c) `

Example:                     f(x) = x4

The integration of f(x) =`int` f(x) dx  = ` x ^5/5`

Integration rule 5:

Integration uv rule:  `int` u dv = uv - `int` v du

Example: find `int` `(x/3) ` cos x dx

Solution :     Put  u  = `x/3` ( first function)   and dv  = cos x dx (second function)

` (du)/dx ` = `1/3`                                v =  sin x

Then, Integration by parts gives,

` int` udv = uv - `int` v du

`int ` `x/3 ` cos x dx = ` x/3` sin x -` int` sinx`(1/3)` dx

= `x/3` sin x - ( -`(1/3)` cos x ) + c

= ` x/3` sin x + `1/3` cos x +  c

Answer :      ` x/3` sin x +`1/3` cos x +  c

I am planning to write more post on what is a differential and tangent function. Keep checking my blog.

Differentiation Rules

Here are the differentiation rules explained with examples:

Differentiation rule 1:

Constant rule:  ` d/dx`` [b f(x)]` = `b ` `d/dx` ` [f(x)] `

Example:                        If f(x) = 6x.  The differentiation of f(x) =6 `d/dx` (x) = 6

Differentiation rule 2:

Differential sum rule :  `d/dx [f(x) + g(x)] = d/dx f(x) + d/dx g(x)`

Example:                     f(x) = x2 + 5

The differentiation of f(x) =f '(x) = 2x

Differentiation rule 3:

Differential Difference rule:   `d/dx [f(x) - g(x)] = d/dx f(x) - d/dx g(x)`

Example:                    f(x) = x3 - 5x

The differentiation of f(x) =f '(x) = 3x2 - 5

Differentiation rule 4:

Differential product rule:   `d/dx [f(x) * g(x)] = f(x) [d/dx g(x)] + g(x) [d/dx f(x)]`

Example:                    f(x) = x2 (x + 10)

The differentiation of f(x) = f '(x) = x2 (1) + (x + 10) 2x

= x2 + 2x2+20x

= 3x2 +20x

Differentiation rule 5:

Differential quotient rule:   `d/dx [f(x)/g(x)] = (g(x) [d/dx f(x)] - f(x) [d/dx g(x)])/[g(x)]^2`

Example:                    f(x) =` x^2/ (x + 4)`

The differentiation of f(x) = f '(x) =  `(((x + 4) 2x) - ((x^2) 1))/(x + 4)^2`

= ` (2x^2+8x - x^2)/(x + 4)^2`

= ` (x^2 + 8x)/(x + 4)^2`

Differentiation rule 6:

Differential power rule:  `d/dx` f(x)n =  n f(x)n-1 `d/dx f(x) `

Example:                    f(x) = x4

The differentiation of f(x) =f '(x) = (4 × 1) x(4 - 1)

= 4 x3

Differentiation rule 7:

Functions of other differential function : `d/dx f[g(x)]`

let u = g(x)           and y = f(u)

So,. `d/dx f[g(x)]` = `dy/(du) f(u) (du)/(dx) g(x)`

Example differentiation problem :

Differentiate the given function:  x2 +  5 tan e2x

Solution:

We consider the second term    5 tan e2x

Let   u = `e^(2x)`   and   y = 5 tan u

`(dy)/(dx)` = `((dy)/(du))` `((du)/(dx))`

So,      `(du)/(dx)` =  `2e^(2x)`

`(dy)/(du)`   = 5sec2 u                                              (derivative of tan u = sec2 u)

`(dy)/(dx)` = `((dy)/(du))`` ((du)/(dx))`

=  5 sec2 u (2e2x)

=  10 sec2 e2x (e2x)

=  10 e2x  sec2 e2x

The derivative of x2 = 2x

So,  The derivative of x2 +  5 tan e2x is 2x + 10 e2x  sec2 e2x

Answer:  The derivative of x2 +  5 tan e2x is 2x + 10 e2x  sec2 e2x

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