Integration by Part
If u and v are two functions of x, then
Remark: The proper choice of u and v is very important to apply integration by part. We can choose the first function (u) as the function which comes first in the word ILATE. Here
I
stands for the inverse trigonometric function (Ex. 
)
L
stands for logarithmic function (Ex. 
)
A
stands for the algebraic functions (Ex. 
, etc.)
T
stands for the trigonometric functions (Ex. 
)
E
stands for the exponential functions. (Ex. 
)
Example 1: Evaluate 
Solution: Here as per the above
remark ILATE 
(Algebraic function) comes first and 
(trigonometric function).
Thus let 
and 
.
Now by formula
Example
2: Evaluate 
Solution: Here as per the above
remark ILATE 
(Algebraic function) comes first and 
(trigonometric function).
Thus let 
and 
.
Now by formula
Where 
and 
Here in second part of integration () we will apply
integration by part again.
Here as per the above remark ILATE (Algebraic function) comes first and 
(trigonometric function).
Thus let and 
.
Now by formula
Thus
Example
3: Evaluate 
Solution: Here as per the above remark ILATE 
(logarithmic function) comes first and 
(Algebraic function).
Thus let 
and 
.
Now by formula
Example
4: Evaluate 
Solution: Given
let 
Putting these values in (1), we get
Now,
as
per the above remark ILATE 
(Algebraic function) comes first and 
(trigonometric function).
Thus let 
and 
.
Now by formula
Example
5: Evaluate 
Solution: Given
Here the formula ILATE does not works.
We know that 
and 
Thus let 
and 
.
Now by formula
Example
6: Evaluate 
Solution: As per the remark ILATE 
(inverse trigonometric function) comes first
and (Algebraic function).
Thus let 
and 
.
Now by formula
Example
7: Evaluate 
Solution: As per the remark ILATE 
(logarithmic function) comes first and (Algebraic function).
Thus let 
and .
Now by formula
Example
8: Evaluate 
Solution: let 
, we get
As per the remark ILATE 
(inverse trigonometric function) comes first
and 
(Algebraic function). Thus let 
and 
.
Now by formula
Example
9: Evaluate 
Solution: let 
Where
