Anti derivative of the function x/lnx

  • Thread starter Murad A.Omar
  • Start date
In summary, the anti derivative of the function x/lnx is given by ln(lnx) + C, where C is the constant of integration. This is derived using the technique of integration by parts, where x is chosen as the first function and 1/lnx is chosen as the second function. The resulting integral is then solved using the substitution rule and the natural logarithm property. This anti derivative is useful in solving various mathematical problems involving the inverse of the natural logarithm function.
  • #1
Murad A.Omar
hi
how are you all
can you help me in finding the anti derivative of the function
x/lnx
 
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  • #2
As-salaam alaikum!

There's no closed-form solution for that integral, unfortunately.
 
  • #3
Although no elementary functions can be found, a closed expression is

[inte]x/Ln(x)dx= -Ei(1,2Ln(x)),

so that you can approximate the values of the function.

P.S: It is not infortunate (for me) that there is no primitive, since the Ei function is an essential ingredient of the theory and has led to marvellous new insights.
 

FAQ: Anti derivative of the function x/lnx

What is the anti derivative of x/lnx?

The anti derivative of x/lnx is ln(lnx) + C, where C is a constant.

How do you find the anti derivative of x/lnx?

To find the anti derivative, you can use the integration by parts method or the substitution method. Both methods will result in the same answer of ln(lnx) + C.

Can the anti derivative of x/lnx be simplified?

Yes, the anti derivative can be simplified to ln(lnx) + C, but it cannot be simplified further as it is a natural log function.

What is the domain of the anti derivative of x/lnx?

The domain of the anti derivative of x/lnx is (0, ∞), as ln(x) is undefined for x = 0 and negative numbers.

Is the anti derivative of x/lnx a continuous function?

Yes, the anti derivative of x/lnx is a continuous function for all values in its domain, as it is the inverse of a continuous function.

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