Question

How do you find the antiderivative of absolute value function \[f\left( x \right) = \left| x \right|\]?

Answer 1

Hint:
In the given question, we have been given a function. We have to find the antiderivative of the given function. The function is the absolute value function. Say \[f\left( x \right)\] is the antiderivative of \[g\left( x \right)\]. Then, it means that when \[f\left( x \right)\] is differentiated, we get \[g\left( x \right)\], or, when we integrate \[g\left( x \right)\], we get \[f\left( x \right)\]. So, we have to find such values, which when differentiated, give us the given function, or simply, we can say that we have to find the result of the integration of the given function.

Formula Used:
To find the antiderivative \[\left( {f\left( x \right)} \right)\] of \[g\left( x \right)\], we have:
\[\int {g\left( x \right)} = f\left( x \right)\]

Complete step by step answer:
The given trigonometric function in the question is \[\sin x\]. To find the antiderivative \[\left( {f\left( x \right)} \right)\] of the absolute value function \[\left( {g\left( x \right)} \right)\], we are going to integrate it,
\[\int {g\left( x \right)} = f\left( x \right)\]
Hence, we have been given \[g\left( x \right)\] which is equal to \[\left| x \right|\] and we have to find the value of \[f\left( x \right)\].
According to the formula,
\[\int {g\left( x \right)} = f\left( x \right)\]
We are going to have to split the value of the function,
If \[x \ge 0\], then \[\left| x \right| = x\] and thus,
\[f\left( x \right) = \int {xdx} = \dfrac{{{x^2}}}{2} + c\]
Then, if \[x \le 0\], then \[\left| x \right| = - x\] and thus,

\[f\left( x \right) = \int {\left( { - x} \right)dx = - \dfrac{{{x^2}}}{2} + c} \]

Note:
In this question, we had to find the antiderivative (or integral) of the given function. This can be calculated only by learning the integrals of the functions by breaking them. There is no method to calculate them. So, it is necessary that we learn the formulae of integrals. This is the most common place where the students make an error - they do not remember the correct integrals, confuse them with something else and get the wrong result. So, it is important to remember the basic integrals of all standard functions.