Question

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

Answer 1

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 which is usually represented as C.
The most general antiderivative of$f(x)$ is $F(x) + C$ where $F(x) = f'(x)$.

Complete step by step solution:
Here the antiderivative has to be found for the function $\dfrac{2}{x}$. The antiderivative of the function is given by the expression$\int {\dfrac{2}{x}} dx$.
To find an antiderivative for a function f, we can often reverse the process of differentiation, add a constant and use the rules for integrals .
The above expression can be written as $\int {2 \times \dfrac{1}{x}} dx$.
It is known to us that, $\int {\dfrac{1}{x}} dx$= $\ln |x| + C$.
A rule for integrals, states that $\int {cxdx = c\int {xdx} }$.
Applying the rule for integrals the expression given to us which is $\int {\dfrac{2}{x}} dx$ becomes,
$\int {2 \times \dfrac{1}{x}} dx = 2\int {\dfrac{1}{x}} dx$
Applying the antiderivative of $\dfrac{1}{x}$we get,
$2\int {\dfrac{1}{x}} dx$ $= 2 \times (\ln |x| + C)$
$= 2\ln |x| + 2C$
$= 2\ln |x| + {C_1}$(Here ${C_1} = 2C$, which is another constant).
Thus, the Anti derivative of $\dfrac{2}{x}$ is $2\ln |x| + {C_1}$.

Note:

An integral generally has a fixed limit, while an antiderivative is more general and will almost always have a $+ C$, the integration constant, at the end. This is the only distinction between the two; otherwise, they are identical.
A mathematical object that can be viewed as an area or a generalisation of an area is called an integral.
The need for integration is critical. Calculating the Centre of Mass, Centre of Gravity, and Mass Moment of Inertia of a Sports Utility Vehicle, for example. To calculate an object's velocity and orbit, estimate planet positions, and comprehend electromagnetism, integration is used.