Question

How do you evaluate the integral $\int{\dfrac{1}{x}}$ from [0,1]?

Answer 1

In this question, we have to find the value of a definite integral. Thus, we will apply the integration formula to get the solution. First, we will rewrite the given integral in the form of$\int\limits_{0}^{t}{{{x}^{-m}}dx}$ . Then, we will apply the integration formula $\int\limits_{0}^{t}{{{x}^{-m}}dx}=\dfrac{{{x}^{-m+1}}}{-m+1}$ in the new integral. After that, we will put the value of limits in the place of x in the answer by using various mathematical rules, to get the required result for the solution.

Complete step by step solution:
According to the problem, we have to find the value of integral.
Thus, we will apply the integration formula and the basic mathematical rules to get the solution.
The integral given to us is $\int{\dfrac{1}{x}}$ from [0,1] ------------- (1)
As we know, the integral of $\dfrac{1}{x}$ is equal to log x, thus applying the same formula in equation (1), we get
$\Rightarrow \left[ \log x \right]_{0}^{1}$
Now, we will apply the definite integral formula $\int\limits_{a}^{b}{f(x)dx}=f(b)-f(a)$ in the above equation, we get
$\Rightarrow \log 1-\log 0$
Now, we know that log (1) is equal to 0, therefore we get
$\Rightarrow 0-\log 0$
Also, we know that log (0) is not defined, thus we get
$\Rightarrow -\infty$ which is not defined.
Therefore, the value of integral $\int{\dfrac{1}{x}}$ from [0,1] is not defined.

Note:
While solving this problem, do mention the formula you are using to avoid confusion and mathematical error. One of the alternative methods to solve this problem is use rewrite the given equation in terms of $\int\limits_{0}^{1}{{{x}^{-1}}dx}$ and then put the limits in the place of x, to get the required result for the solution. That is, we get the value of integration as not defined.