Question

How do you evaluate the integral $\int {\dfrac{{dt}}{{t\ln t}}} $ ?

Answer 1

Hint: In the given question, we have been asked to integrate the following numerical. In order to solve the question, we integrate the numerical by following the substitution method. We replace $u = \ln t$ and solve the numerical by further integration. After we have simplified our sum, we just need to integrate the terms and then we replace the substituted variables by the original variables.

Complete step by step solution:
We have the given function,
$\int {\dfrac{{dt}}{{t\ln t}}} $
let $u = \ln t$, and differentiate this expression , we will get the following ,
$du = \dfrac{1}{t}dt$
Now, substitute the above value into the given expression, we will get the following result,
$ \Rightarrow \int {\dfrac{{du}}{u}} $
Now solving,
$ \Rightarrow \int {\dfrac{{du}}{u}} $
This is the standard integral which is equal to $\ln (u)$ .
Now, replace the value of $u = \ln t$ ,
We will get ,
$ \Rightarrow \int {\dfrac{{du}}{u}} = \ln (|\ln t|) + c$
Here, we need to remember that we have to put the constant term $c$ after the integration equation and the value of the given constant i.e. $c$ .
Hence, we get the required result.

Note: Here, we need to remember that we have to put the constant term C after the integration equation and the value of the given constant i.e. C, it can be any value 0 equal to zero also. In order to solve the question that is given above, students need to know the basic formula of integration and they should very well keep all the standard integral into their mind because sometimes the given integration is the standard integral and we do not need to solve the question further and directly write the resultant integral.