Question

What is the antiderivative of $\dfrac{3}{x}$ ?

Answer 1

Hint : In calculus, the anti derivative of a function is the same as the indefinite integral of the given function. The given question requires us to integrate a function of x with respect to x. Integration gives us a family of curves. Integrals in math are used to find many useful quantities such as areas, volumes, displacement, etc. integral is always found with respect to some variable, which in this case is x.

Complete step by step solution:
The given question requires us to integrate a rational function $\dfrac{3}{x}$ in variable x whose numerator is $3$ and whose denominator is $x$ . So, we first represent the function in negative power form and then integrate the function directly using the power rule of integration.
So, we can write $\dfrac{3}{x}$ as $3{x^{ - 1}}$ .
Hence, we have to integrate $3{x^{ - 1}}$ with respect to x.
So, we have to evaluate $\int {3{x^{ - 1}}} dx$ .
Now, we can take the constant $3$ outside of the integral. So, we get,
$\Rightarrow \int {\dfrac{3}{x}\,} dx = \int {3{x^{ - 1}}} dx = 3\int {{x^{ - 1}}} dx$
Now, we know that the integral of $\dfrac{1}{x}$ is $\ln x$ .
$\Rightarrow \int {\dfrac{3}{x}\,} dx = 3\ln x + c$
Adding the arbitrary constant of indefinite integration, we get the value of anti-derivative of $\dfrac{3}{x}$ as $\left( {3\ln x + c} \right)$ .
So, the correct answer is “ $\left( {3\ln x + c} \right)$ ”.

Note : 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 constant.