Thursday, 23 April 2015
On 23:12 by Vardan Kumar in Mathematics 2 comments
INTEGRATION BY PARTSEASY METHOD WITH EXAMPLES
Integration by parts as we all know is the integral part of Mathematical Calculus.There are various formulas and tricks associated with integration of functions.Of all the tricks and formulas "Integration By Parts" is one of the crucial method or formula used to find the integration or antiderivative of product of functions more specifically two functions.
PRINCIPLE:
The logic behind this type of integration is,it frequently decomposes the integration of product of two functions into an integration problem which is easily solvable.
Mathematical Formula:
Let us suppose we have two functions u and v,we need to find integration of uv then......
∫uv dx=u∫vdx∫u'(∫vdx)dx
Here u>Any function u(x)
v>Any function v(x)
u'>Derivative of u(x)
Using this formula may cause confusion and increases the chances of errors.Also we need to memorize this formula.So here we go with....
EASY METHOD:
Before we begin with the discussion lets just have a quick review of ILATE rule.Using this rule we are gonna characterize our function as Function 1 and Function 2.
IINVERSE TRIGONOMETRIC FUNCTIONS
LLOGARITHMIC FUNCTIONS
AALGEBRAIC FUNCTIONS
TTRIGONOMETRIC FUNCTIONS
EEXPONENTIAL FUNCTIONS
The above order is used in the following way:
The function which appears above the another function in the above list is named as Function 1 else Function 2.
Integration By PartsEasy Method 
STEPS TO BE FOLLOWED:
1.Characterize the functions as Function 1 and Function 2.
2.Form two columns with heading Function 1 and function 2 as shown in the above image.
3.Carry out continues differentiation and integration on function 1 and function 2 respectively till 'Stop' is obtained.
4.Draw arrows as shown in the above image.
5.The inclined arrow indicates the simple multiplication.
6.The linear arrow indicate to multiply the two functions and then perform integration on the multiplication obtained.
Note:START WITH '+' SIGN IN MULTIPLICATION AND ALTER SIGN IN EACH STEP.
Let us illustrate this method with Examples.
EXAMPLE 1:∫xcosx dx
Solution:Open the below image for solution
IntegrationExample 1 
LET US TAKE ANOTHER EXAMPLE.......
EXAMPLE 2: ∫e^x.cosx dx
Solution:Open image below for solution
IntegrationExample 2 
LET US REVIEW ONE LAST EXAMPLE.........
EXAMPLE 3:∫x.logx dx
Sol:Refer the image below for solution.....
`
IntegrationExample 3 
NOTE:THE CONSTANT OF INTEGRATION IS NOT INTRODUCED HERE.IT DOES NOT MEAN THAT IT WON'T COME.THE CONSTANY OF INTEGRATION IS ALWAYS INTRODUCED AFTER EVALUATING THE INDEFINITE INTEGRAL.
This was Integration by partsEASY METHOD.For any queries feel free to comment below or refer the contact form below.
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I loved it, integration is so simple now......
ReplyDeleteThats good....if you have any queries feel free to ask.
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