Integration as Inverse Process of Differentiation: Definition, Theorem, Problems & Class 12 Maths Notes

Integration can be understood as the inverse operation of differentiation. Differentiation finds the rate of change of a function, while integration finds the function given its rate of change (or its derivative). Essentially, if you differentiate a function and then integrate the result, you will get back to the original function (plus a constant in the case of indefinite integrals). Integration is the inverse process of differentiation. Thus, finding integrals (antiderivatives) of a function f(x) means finding a function F(x). whose derivative is f(x).
If F’(x) = f(x) i.e., if $\frac{d}{dx}\left(F\right(x\left)\right)=f\left(x\right)$ , then we write ∫ 𝑓(𝑥)𝑑𝑥 = F(x). The symbol ∫ is called an integral sign here; dx denotes the differential of x. The symbol ∫ 𝑓(𝑥)𝑑𝑥 is read as “integral of f(x) with respect to x”.
In simpler terms:
a) Think of differentiation as breaking a whole into its parts (like taking a cake and finding the amount of each ingredient.
b) Think of integration as putting those parts back together to form the whole (like taking the amounts of the ingredients and reconstructing the cake.

Important Topics:

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Differentiation:

Differentiation is a process of finding the derivative of a function. It tells us how a function is changing at any point. 
Integration:

It is a process of finding the antiderivative of a function.  
Constant Integration:

While finding the antiderivative, we need to add a constant of integration, because the derivative of a constant is always zero. This means that many functions can have the same derivative, and they all differ by a constant. Therefore, when you integrate, you could have any constant added to the result, and it would still have the same derivative.

This inverse relationship is fundamental to calculus and has many important applications in various fields like physics, engineering, and economics.

Here, we have mentioned the difference between differentiation and integration.

Both operations are defined on functions

All functions are not differentiable. Similarly, all functions are not integrable.

Both are linear operators as

$\begin{array}{c}\left(i\right)\frac{d}{dx}\left[K_1{f}_1\right(x)+K_2{f}_2(x\left)\right]=K_1\frac{d}{dx}{f}_1\left(x\right)+K_2\frac{d}{dx}{f}_2\left(x\right)\\ \left(ii\right)\int \left[K_1{f}_1\right(x)+K_2{f}_2(x\left)\right]dx=K_1\int f_1\left(x\right)dx+K_2\int f_2\left(x\right)dx\end{array}$

The derivative of a function is unique, whereas the integral of a function is not unique. Different integrals of a function differ by a constant.

We can discuss the derivative of a function at a point, but we never speak of the integrals at a point. We discuss the integral of a function on an interval.

The geometric derivative of a function at a point represents the slope of the tangent at that point. Similarly, the indefinite integral geometrically represents a family of curves placed parallel to each other, having parallel tangents at that point of intersection of the curves of the family with the lines orthogonal to the axis representing the variable of integration.

Both differentiation and integration involve limits.