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LORD of D Last Bench

Remember ILATE if u don't I'll tell you it's the sequence of preference of integration when more than one term is provided. The alphabets stand for the following

I- Inverse

L- Logarithimic

A-Algebraic

T- Trignometrical

E- Exponential

This rule is used to identify the function in the higher priority

Example

Question: x logx

The Usual method

Int[x logx] = : logx Int[x] - Int[{diff. Logx} Int[x]]

X²/2logx - Int[1/x . X²/2]

X²/2logx -x2/4 + C

Note Int stands for Integration sign and the end of the bracket indicate dx

The shortcut

1.Make two columns write diff and int above each one

2. Below the diff column write the more preferred function and the left below the int column

3. Now differentiate till a linear, term or zero is obtained. And integrate the function same number of times as you have differentiated.

I will take the above question to Explain the trick

Diff Int

logx x

1/x x²/2

x3/4

Mutliply the element of diff column with the next element of int column. And do this till the diff column ends. If the even element of diff column is selected then you need to multiply by -1 too.

Ans. = x²/2 logx -x²/4 +C

In the above question i have stopped differentiating beacuse I had obtained a term of first degree.

This process is very effective while attempting questions in which there is a cgoice for the answers.