Question

Solve $\int {\dfrac{{3x}}{{3x - 1}}dx}$

Answer 1

Hint : The equation in the question is the simple type of integration that can be solved using some simple operation.
We can solve it by adding a term or number in the both numerator or denominator.
As well as we can solve it by rationalizing the denominator method.

Complete step by step solution:
Let us take the equation,
$\Rightarrow$ $\int {\dfrac{{3x}}{{3x - 1}}dx}$
By adding and subtracting $1$ in both numerator and denominator,
$\Rightarrow$ $\int {\dfrac{{\left( {3x - 1} \right) + 1}}{{3x - 1}}}$
$\Rightarrow \int {1dx + \int {\dfrac{{dx}}{{3x - 1}}} }$
By integrating and using logarithmic integration,
$\Rightarrow x + \dfrac{1}{3}\log (3x - 1) + c$
So, the correct answer is “ $x + \dfrac{1}{3}\log (3x - 1) + c$ ”.

Note :

Integration is the calculation of an integral.
The integration denotes the summation of discrete data. The integral is calculated to find the functions which will describe the area, displacement, volume, that occurs due to a collection of small data, which cannot be measured singularly.