Question

What is the antiderivative of $\dfrac{1}{{2x}}$ ?

Answer 1

Hint: A differentiable function ${\text{F}}$ whose derivative is equal to the original function ${\text{f}}$ is known as an antiderivative of a function ${\text{f}}$ in calculus. It can be expressed as $F’ = f$.

Complete step-by-step solution:
The antiderivative is the name we assign to the procedure that goes backward from a function's derivative to the function itself (rarely). Since the derivative does not fully define the function (you may add any constant to the function and the derivative will remain the same), you must have more information to return to an explicit function as an antiderivative.
Let $f\left( x \right)\, = \,\dfrac{1}{{2x}}$
We can find $F\left( x \right)$ by calculating the derivative's infinite integral.
$F\left( x \right)\, = \,\int {f(x)dx} $
On substituting the value of $f\left( x \right)$ we get,
$F\left( x \right)\, = \,\int {\dfrac{1}{{2x}}dx} $
Move $\dfrac{1}{2}$ out of the integral because it is a constant with respect to $x$.
$F\left( x \right)\, = \,\dfrac{1}{2}\int {\dfrac{1}{x}dx} $
$\ln \left( {|x|} \right)$is the integral of $\dfrac{1}{x}$with respect to $x$
\[\therefore \,F\left( x \right)\, = \,\dfrac{1}{2}\left( {\ln \left( {|x|} \right)\, + \,C} \right)\]
$\dfrac{1}{2}\left( {\ln \left( {|x|} \right)\, + \,C} \right)$ is the antiderivative of the function $f\left( x \right)\, = \,\dfrac{1}{{2x}}$
Additional Information:
While ancient Greek mathematics provided methods for measuring areas and volumes, the concepts of integration were developed independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late $17th$ century, who thought of the area under a curve as an infinite sum of rectangles of infinite width.
Antiderivatives are derivatives' polar opposites. An antiderivative is a function that does the opposite of the derivative. Many antiderivatives exist for a single function, but they all take the form of a function plus an arbitrary constant. Indefinite integrals are incomplete without antiderivatives.

Note: The indefinite integral symbol written is used to describe all antiderivatives of a function $f\left( x \right)$, where, the function $f\left( x \right)$ is called the integrand, and the constant of integration is referred to as $C$.