Question

What is \[\int {\dfrac{{\ln x}}{x}} dx\] equal to?
1. \[\dfrac{{{{\left( {\ln x} \right)}^2}}}{2} + c\] where \[c\]is the constant of integration
2. \[\dfrac{{\left( {\ln x} \right)}}{2} + c\] where \[c\]is the constant of integration
3. \[{\left( {\ln x} \right)^2} + c\] where \[c\]is the constant of integration
4. None of the above

Answer 1

Hint: The given question requires us to integrate a function of x with respect to x. We evaluate the given integral using the substitution method. We substitute the logarithmic function as a new variable and then convert the integral in respect to the new variable which is easier to solve.

Complete step-by-step solution:
 “Integral is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs.”
We know that differentiation is the process of finding the derivative of the functions and integration is the process of finding the antiderivative of a function. So, these processes are inverse of each other. The integration is also called the anti-differentiation.
Two types of integrals:
1. Definite Integral
2. Indefinite Integral
Definite Integral
An integral that contains the upper and lower limits then it is a definite integral. On a real line, x is restricted to lie. Riemann Integral is the other name of the Definite Integral.
A definite Integral is represented as:
\[\int\limits_a^b {f(x)dx} \]
Indefinite Integral
Indefinite integrals are defined without upper and lower limits. It is represented as:
\[\int {f(x)dx} = F(x) + c\]
Where \[c\] is any constant and the function \[f(x)\] is called the integrand.
Put \[\ln x = t\]
Differentiating both sides we get \[\dfrac{1}{x}dx = dt\]
Therefore \[\int {\dfrac{{\ln x}}{x}} dx = \int {tdt} \]
Since \[\int {tdt} = \dfrac{{{t^2}}}{2} + c\]
Therefore we get \[\int {\dfrac{{\ln x}}{x}} dx = \dfrac{{{{\left( {\ln x} \right)}^2}}}{2} + c\] where \[c\]is the constant of integration.
Therefore option (1) is the correct answer.

Note: The integration denotes the summation of discrete data. The indefinite integrals of certain functions may have more than one answer in different forms. However, all these forms are correct and interchangeable into one another. Indefinite integral gives us the family of curves as we don’t know the exact value of the arbitrary constant.