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How to evaluate the definite integral of lnx from 0 to 1?

Answer
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Hint: To solve this question we need to know the concept of definite integral. We are given with the limits of the integral for the given function. For integrating the given function,lnx the question will be solved using the method of integration by parts. So the formula used here will be udv=uvvdu, where u and v are the two functions.

Complete step by step answer:
To solve the integration of the function we should know the formula of the integration. The question is lnxdx with limit as [0,1], which could be written as
01lnxdx
The formula used to integrate the question lnx is udv=uvvdu is , if the function is lnx is u and the function dv=dx. Mathematically it will be written as:
01lnxdx
On applying the formula we get intudv=uvvdu , where u=lnx on differentiating both side of the function lnx we get:
du=1xdx
Also the term dv is considered as dx, which means:
dv=dx
The above equation on integrating both side becomes:
v=x
Considering the formula the equation becomes:
uvvdu
On substituting the values on the above formula we get:
[xlnx]0101dx
Integration of dx is x.
[xlnxx]01
On substituting the values on the function with the limits we get:
[1×ln11][0×ln00]
The value of ln1=0 and ln0 is not defined. Putting the values in the above expression, we get:
[01][00]
On further calculation it becomes:
1

The integration of the function lnx from [0,1] is 1.

Note: To solve this question we need to remember the formulas for integration, so that we do not make mistakes while writing the formulas. The above question has been solved by the method of integration by parts. The concept of definite integral is very useful and has a large number of applications. It is used to find the area of the enclosed figure etc.
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