Integration as the inverse of differentiation

Introduction

Integration as inverse of differentiation is the process of evaluating an indefinite integral or a definite integral. The indefinite integral of a function f is defined as a function g with a derivative d[g(x)]/dx =f(x). It is important to note that integration is the opposite process of differentiation. Rather than differentiating a function, you are given its derivative and asked to determine its original function – referred to as integration or antidifferentiation.

We employ integration as the inverse of differentiation to get integrals and antiderivatives. As we utilise the fundamental theorem of calculus, we must determine the antiderivatives.

What is antidifferention?

Antidifferentiation is the process of determining a function’s antiderivative, much as differentiation is the process of calculating a function’s derivative.

What are antiderivatives?

Antiderivatives are the inverse of derivatives. An antiderivative function produces the opposite of what a derivative does. Many antiderivatives exist for a single function, but they always have a function plus an arbitrary constant. Indefinite integrals rely heavily on antiderivatives.

Finding a function’s antiderivatives takes some reverse reasoning. Because the provided function is derivative of the desired antiderivative, verifying correctness is simple – take the derivative and check to see whether it is the supplied function. Furthermore, antiderivatives of functions aren’t just one function but an entire family of functions. This family is denoted by a polynomial plus c, where c might be any constant.

Formula for antiderivatives:

Now that we’ve established the physical foundation for the antiderivative, we can unveil the formula to compute them. Using this formula to get the antiderivative of a function is simple since you don’t have to worry about how the function’s graph appears. And it’s fairly straightforward for non-trigonometric functions.

For functions, f(x)=axn the antiderivative is equal to,

F(x) = 1n+1axn+1 + C 

Many of these terms may be unfamiliar to you. In mathematics, f(x) is a general term for a function. The a stands for any constant integer, while the n represents an exponent. The symbol C is equal to a constant, but it performs a distinct function. Finally, F(x) denotes the antiderivative of function f(x).

Antiderivatives rules:

These are fundamental principles for determining the antiderivatives of various function combinations in calculus. They determine the antiderivative of a function’s sum or difference, product, quotient, scalar multiple, and constant. These principles may be used for antidifferentiation of algebraic, exponential, trigonometric, hyperbolic, logarithmic, and constant functions.

Basic antiderivative rules:

Here are the rules for antidifferentiation algebraic functions having power and other function combinations:

Antiderivative chain rule:

Since antidifferentiation is the opposite of differentiation, the rules of derivatives lead to specific antiderivative rules like the antiderivative chain rule (or the u-substitution technique of antidifferentiation). If the integral is of the form ∫u’ (x) f(u(x)) dx, the antiderivative chain rule is applied.

Antiderivative power rule:

∫xndx=xn+1/(n+1)+C, where n ≠ -1, is usually referred to as the antiderivative power rule.

Antiderivative product rule:

The antiderivative product rule is often referred to as the integration as the inverse of differentiation by parts approach. It is one of the significant antiderivative rules for determining the antidifferentiation of the product of functions. The antiderivative product rule has the formula ∫(uv)dx=u∫v.dx-∫[u.∫v.dx]dx+ c.

Antiderivative quotient rule:

The antiderivative quotient rule applies when the function is presented as numerator and denominator. Another way to determine the antiderivative of functions’ quotients is to consider a function of the form f(x)/g(x). If f(x) = u and g(x) = v, then the antiderivative quotient rule is

∫du/v=u/v+∫[u/v2]dv.

Antiderivative rule for scalar multiple of function:

We may use the formula ∫kf(x) dx = k∫ f(x) dx to get the antiderivative of a scalar multiple of a function f(x). It means that the antidifferentiation of kf(x) is equal to k times the antidifferentiation of f(x), in which k is a scalar.

Important antiderivative rules notes:

• The antiderivatives rules are used to calculate the antiderivatives of various algebraic, trigonometric, logarithmic, exponential, inverse trigonometric, and hyperbolic function combinations.
• Most differentiation rules have matching antiderivative rules for antidifferentiation.
• For a constant function f(x) = k, the antiderivative rule is ∫kdx=kx+C.

Conclusion:

Integration as the inverse of differentiation is essentially reverse differentiation; that is, we start with a given function f(x) and ask which functions, F(x), have f(x) as their derivative. The resulting quantity is known as an indefinite integral. By replacing values with the indefinite integral, a definite integral may be derived. Integration is a kind of inverse differentiation procedure. Instead of dividing a function, we are given its derivative and asked to discover its primitive or original function. This is referred to as integration as the inverse of differentiation or antidifferentiation.