Question

How do you find the integration of log x?

Answer 1

Hint: The question provided to us is pretty simple and tricky. You need to use the technique of integration by parts. After using the formula of integration by parts, you need to simplify the equation, and hence, you will get the result. So, let’s see what will be the approach.

Complete Step by Step Solution:
The given problem statement is to find the integration of log x.
The first thing which we need to do is the use of integration sign, that means,
$\int \log xdx$
Now, we need to use the formula for integration by parts,
$\Rightarrow \int f(x)g(x)dx=f(x)\int g(x)dx-\int (f^{\prime }x\int g(x)dx)dx$
So, here we will assume that the first function will be our log x and the second function will be the constant 1, that means,
$\Rightarrow f(x)=\log x$ and $g(x)=1$
By putting the required values in the formula, we get,
$\Rightarrow \int (\log x).1dx=\log x\int 1.dx-\int ({\displaystyle \frac{d(\log x)}{dx}}\int 1.dx)dx$
Now, when we solve, we know the integration of dx is x and the differentiation of log x is ${\displaystyle \frac{1}{x}}$, so we get,
$\Rightarrow \int (\log x).1dx=(\log x)x-\int {\displaystyle \frac{1}{x}}.xdx$
Now, x in the numerator and denominator will cancel out and, again the integration of dx will be x, so we get,
$\Rightarrow \int (\log x).1dx=x\log x-x+c$

Therefore, the integration of log x is $x\log x-x+c$

Note:
Whenever you are solving sums or problems in the domain of integration, you need to have a good grasp of the formulas related to integration as well as differentiation. Here, in the above question, we have done integration with dx as well as differentiation of log x.