Question

How do you integrate ln(3x)?

Answer 1

We will first write the fact that $\int {\ln (nx)dx = } x\ln (nx) - x + C$. Now, we will just put in n = 3 and thus, we will get the required solution to the given question.

Complete Step by Step Solution:
We are given that we are required to integrate ln (3x).
We will first find the integration of ln (nx).
We can write this function as multiple of 1 and ln (nx).
Now, we need to find the value of $\int {1.\ln (nx)dx}$.
Now using the ILATE rule, we have ln (nx) as the first function and 1 as the second function.
Now, we get:-
$\Rightarrow \int {1.\ln (nx)dx} = x\ln (nx) - \int {\dfrac{n}{{nx}}xdx + C}$
We can write this as follows:-
$\Rightarrow \int {\ln (nx)dx} = x\ln (nx) - \int {dx + C}$
Now integrating dx on the right hand side in above equation, we will then obtain the following equation:-
$\Rightarrow \int {\ln (nx)dx} = x\ln (nx) - x + C$
Now, putting in n = 3, we will then obtain the following equation with us:-

$\Rightarrow \int {\ln (3x)dx = } x\ln (3x) - x + C$
Thus, we have the required answer.

Note:
The students must know the ILATE rule which has been mentioned above.
I stands for Inverse, L stands for Logarithmic, A stands for Algebraic, T stands for Trigonometric and E stands for exponential. This is the sequence in which we take the first function. Therefore, we have taken ln (3x) as the first function and 1 as the second function.
Also, note that if there are two function; first function being f (x) and the second function being g (x).
The integration of f (x) and g (x) is given by the following expression:-
\Rightarrow \int {f(x).g(x)dx = f(x)\int {g(x)dx - \int {\left\\{ {\left( {\dfrac{d}{{dx}}f(x)} \right)\int {g(x)dx} } \right\\}dx + C} } }
This is the formula that we used above.
The students must note that we may also use the formula directly.
We know that we have a formula for integration of ln (nx) which is given by the following expression:-
$\Rightarrow \int {\ln (nx)dx = } x\ln (nx) - x + C$
Now, putting in n = 3, we will then obtain the following equation with us:-
$\Rightarrow \int {\ln (3x)dx = } x\ln (3x) - x + C$
Thus, we have the required answer.