Integration as an Inverse Process of Differentiation

Integration is the process of determining whether an indeterminate or definite integral should be evaluated. A function g with a derivative Dx [g(x)] =f is an indefinite integral (x). It’s worth noting that integration is an inverse process of differentiation. We are given the derivative of a function and asked to discover its original function rather than differentiating it. Integration or antidifferentiation is the term for this process. We are given the derivative of a function and asked to discover its primitive rather than differentiating it. Integration or anti-differentiation is the term for this type of process.

 Let f(x) be a function.  The assortment of primitives is named as the indefinite integral of f(x) and implied by ʃ f(x) dx.

Therefore, d[ф(x) + C]/dx = f(x) => ʃ f(x) dx = ф(x) + C

Where ф(x) is primitive of f(x) and C is an arbitrary constant recognized as the integration constant.

Indefinite integral properties

(i) The integration and differentiation

are one another’s inverses that gives the following results :

d[ʃ f(x) dx]/dx = f(x)

Next,  ʃ f’(x) dx = f(x) + C,

C refers to an arbitrary constant.

(ii) 2 indefinite integrals having the similar derivative have the same curve families; hence they are comparable.

i.e. d[ʃ f(x) dx]/dx = d[ʃ g(x) dx]/dx

=> d[ʃ f(x) dx – ʃ g(x) dx]/dx = 0

=> ʃ f(x) dx – ʃ g(x) dx = C,

 C denotes a real number

=> ʃ f(x) dx = ʃ g(x) dx + C

The arbitrary constant c is referred to as a constant of integration since its derivative is always 0. There may be a constant involved in integration, but it is unknown unless the lower and upper limits of integration are provided for a specific situation. As a result, the unknown value is represented by C. You will be given a task to solve for C that will give you the y(0) value. After that, you can substitute 0 for x and y(0) for y.

On the interval [a,b], f(x) = c, where c is a constant.

Geometrical Interpretation of Indefinite Integral:

∫f(x) dx = F(x) + C = y (say).

A family of curves is represented by y = F(x) + C. We can acquire various members of the family by changing C’s value. Any of the curves may be made parallel to itself to generate these members.

Integration is the Inverse of Differentiation

Finding an integral is the opposite of finding a derivative.

For example, let us find the integral of 2x.

2x=x²

Here, we know that the derivative of x² is 2x, and the integral is x².

Different Types of Integrals in Mathematics-

Till now, we have learned what Integration is. In mathematics, there are two types of Integrations or integrals –

• Definite Integral

• Indefinite Integral

What is a Definite Integral?

The top and lower boundaries of an integral are both contained in a definite integral. The Riemann Integral is another name for the Definite Integral.

•  Representation of a Definite Integral-

∫baf(x)dx

What is an Indefinite Integral?

An indefinite integral is one that lacks upper and lower bounds. Anti-Derivative or Primitive Integral are other names for Indefinite Integral. A differentiable function F whose derivative is identical to the original function f is the indefinite integral of a function f.

Five Different Types of Integration Techniques-

Here’s a list of Integration Methods –

• Substitutional Integration
• Parts Integration Partial Fraction Integration
• Integration of a certain proportion
• Using Trigonometric Identities for Integration

Integration by Substitution:

Let I = ∫f(x) dx. This integral can be transformed into another form by changing the independent variable x to t by putting x=g(t).

∴ dxdt = g'(t) or dx = g'(t) dt

∴ I = ∫f[g(t)]g'(t) dt

Note: While making a substitution, it should be kept in mind that the f[g(t)] is in the form of some standard formula, whereas g'(t) is a factor, along with f[g(t)] e.g.

Consider the integral I = ∫x2 cos(x³ + 2)dx

If we put x³ + 2 = t, its derivative 3x² is a factor and cost can easily be integrated.

Integration by Parts:

For a given functions f(x) and q(x), we have

∫[f(x) q(x)] dx = f(x)∫g(x)dx – ∫{f'(x) ∫g(x)dx} dx

Here, we can choose the first function according to its position in ILATE, where

I = Inverse trigonometric function

L = Logarithmic function

A = Algebraic function

T = Trigonometric function

E = Exponential function

[the function which comes first in ILATE should taken as first junction and other as second function]

Formulas Related to Integration by Parts

∫ ex(f(x) + f'(x)).dx = exf(x) + C

∫√(x² + a²).dx = ½ . x.√((x² + a²)+ a2/2. log|x + √(x² + a²)| C

∫√(x² – a²).dx =½ . x.√(x² – a²) – a²/2. log|x +√(x² – a²) | C

∫√(a² – x²).dx = ½ . x.√(a² – x²) + a² /2. sin-1 x/a + C

IMPORTANT RULES

(i) Rule to integrate ∫ sinm x cosn x dx.

(a) If the index of sin x is a positive odd integer, put cos x = t.

(b) If the index of cos x is a positive odd integer, put sin x = t.

(ii) Rule to integrate : ∫1asin2x+bcos2xdx,∫1a+bcos2xdx ; etc.

a sin x + b cos x J a + b cos x

(a) Divide the numerator and denominator by cos2x

(b) Replace sec2x, if any, in the denominator by 1 + tan2x

(c) Put tan x = t so that sec2x dx = dt.

(iii) Integration of Parts.

Integral of the product of two functions = First function x Integral of second – Integral [(diff. coeff. of first) x (integral of a second)].

Conclusion

As a result, the definite integral, also known as the limit of a Riemann sum, is a number whose value is determined by the function f and the values a and b. An arbitrary constant is used in the indefinite integral.