Question

What is the integral of an integral?

Answer 1

Hint: We know that integration is nothing but the reverse process of differentiation. Similarly, we can define integral of an integral, or double integral, as the reverse process of double differentiation or second derivative or $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}$ .

Complete step-by-step solution:
We know that integration is used to find areas, volumes, central points and many other useful things. We also know that integration is the reverse process of differentiation.
We can represent an integral as per the following, $\int{f\left( x \right)dx}$, where f(x) is the function to be integrated along the variable x.
In the same way, we can say that an integral is the reverse process of the second derivative of any function. We define integral of an integral as double integral.
Integral of an integral is represented as $\int{\int{f\left( x \right)dxdx}}$, where f(x) is the function integrated twice.
Let us understand this with the help of an example.
Let us assume a function f(x) = $3{{x}^{2}}$.
So, we know that when this function is differentiated, we get
$f'\left( x \right)=\dfrac{df\left( x \right)}{dx}=6x$
Let us differentiate f(x) one more time. Now, we get
$f''\left( x \right)=\dfrac{{{d}^{2}}f\left( x \right)}{d{{x}^{2}}}=6$
So, now if we assume a function h(x) = 6, then on calculating the integral of an integral of h(x), we must get a function of the form f(x).
So, let us first calculate the integral of h(x),
$\int{h\left( x \right)dx=6x+C}$ , where C is a constant.
Now, we are integrating the above equation one more time to get an integral. So, we now have
$\int{\int{h\left( x \right)dxdx=3{{x}^{2}}+Cx+D}}$ , where D is another constant.
Here, we can clearly see that, $\int{\int{h\left( x \right)dxdx}}=f\left( x \right)$ , with some constants of integration.
This is how we can define an integral.

Note: We must always remember to write the constant of integration in the case of indefinite integrals. Same as double integral, we can define triple integral and so on, which are collectively known as multiple integrals.