An integral is a way to add up very tiny pieces to find the total of something. It is used to answer questions like:
- How much area is under a curve?
- How far has something travelled if its speed keeps changing?
- How much total quantity is accumulated over time?
Formal Meaning
In mathematics, integration is defined as the inverse operation of differentiation.
- If differentiation breaks things down into rates and small changes,
- then integration puts them back together again.
If the derivative of a function F(x) is f(x), then:
∫f(x) dx = F(x) + C
Here C is the constant of integration.
So integrals are also called antiderivatives.
Example: Given function f(x) = 2x3 + 3x
Derivative of f(x) = f'(x) = d/dx{2x3 + 3x}
f'(x) = 6x + 3
Integration of f'(x) = ∫ f'(x) dx = f(x)
f(x) = 2x3 + 3x + c
Types of Integrals
Integrals are used to solve various types of problems in Calculus, Physics, etc. The integrals are of two types, that are,
- Indefinite Integrals
- Definite Integrals
- Improper Integrals
Indefinite Integrals
Indefinite Integrals are used to find the integrals of the function when the limit of the integration is given. While solving the indefinite integrals we always have the constant of integration in the solution. The integration of the function g(x) is calculated as,
∫ g(x) = G(x) + c
Definite Integrals
Definite Integrals is the integral of the function with the limit of the integration given. Definite integrals gives the value of the function in numerical form. The definite integral of the function is given as,
∫ab g(x) dx = G(b) - G(a)
Improper Integrals
Improper integrals arise when the function being integrated is unbounded or has infinite discontinuities within the interval of integration. They are evaluated by considering limits as one or both of the integration limits approach infinity or approach points of discontinuity within the interval.
Here's a more detailed explanation along with examples:
Infinite Intervals: An improper integral with an infinite interval occurs when one or both of the integration limits are infinite.
Example: Consider the function f(x)=1/x2 over the interval [1, ∞).
∫1∞ 1/x2 dx
This integral represents the area under the curve f(x) from x=1 to x=∞. To evaluate it, we compute the limit:
limt → ∞ ∫1t 1/x2 dx
If the limit exists, the integral is said to converge; otherwise, it diverges.
Infinite Discontinuities: An improper integral with infinite discontinuities occurs when the function has a vertical asymptote or an infinite discontinuity within the interval of integration.
Example: Consider the function g(x)= 1/√x over the interval [0, 1].
∫01 1/√x dx
The function g(x) has a vertical asymptote at x=0. To evaluate the integral, we compute the limit:
lima → 0+ ∫a1 1/√x dx
If the limit exists, the integral converges; otherwise, it diverges.
Functions with Infinite Integrals: Some functions have integrals that extend to infinity due to their behavior.
Example: The function h(x)=1/x over the interval [0, 1].
∫01 1/x dx
This integral is improper because it becomes infinite at x=0. To evaluate it, we compute the limit:
lima → 0+∫a1 1/x dx
If the limit exists, the integral converges; otherwise, it diverges.
Integrals as Inverse of Differentiation
Consider derivatives of some functions given below,
- d/dx {x2} = 2x
- d/dx {sin (x)} = cos x
- d/dx {cos (x)} = - sin x
In the above equations, consider d/dx {x2} = 2x , here x2 is the anti-derivative for the function f'(x) = 2x. This follows for the above derivatives too.
Note that if these functions are added with constants, there is no difference to the derivatives as the derivative of a constant is zero.
- d/dx {x2 + C} = 2x
- d/dx {sin (x) + C} = cos x
- d/dx {cos (x) + C} = - sin x
So, it can be concluded that for any function, it's anti-derivatives are infinite. For example, for the function f(x), let it's anti-derivative be F(x),
∫ f(x) dx = F(x) + C
∫ denotes the integration. This will represent the integration operation over any function. The table below represents the symbols and meanings related to integrals.
| Symbols | Meaning |
|---|---|
| ∫ f(x) dx | Integral of f with respect to x |
| f(x) in ∫ f(x) dx | Integrand |
| x in ∫ f(x) dx | Variable of integration |
| Integral of f(x) | A function such that F'(x) = f(x) |
Properties of Integrals
Linearity: The integral of a sum is the sum of the integrals. In other words, if f(x) and g(x) are integrable functions and a and b are constants, then:
∫ [af(x)+bg(x)] dx = a∫f(x)dx + b∫g(x)dx
Additivity: The integral over a union of disjoint sets is the sum of the integrals over the individual sets.
Constant Multiple Rule: If c is a constant, then:
∫c⋅f(x)dx = c⋅∫f(x)dx
Integration by Parts: Integration by parts is a technique for integrating the product of two functions. The formula states:
∫udv = uv - ∫vdu
Change of Variable: Also known as substitution, this technique involves changing the variable of integration. If u = g(x) is differentiable and f is continuous, then:
∫ f(g(x)) ⋅ g′(x)dx = ∫f(u)du
Integration of Odd or Even Functions: The integral of an odd function over a symmetric interval is zero. The integral of an even function over a symmetric interval is twice the integral over half the interval.
Fundamental Theorem of Calculus: This theorem establishes the relationship between differentiation and integration. It states that if f(x) is continuous on [a, b] and F(x) is an antiderivative of f(x), then:
∫ab f(x)dx = F(b) − F(a)
Symmetry: If a function f(x) is even (symmetric about the y-axis), then its integral over the interval [−a, a] is twice the integral over the interval [0, a]. If f(x) is odd (antisymmetric about the origin), the integral over [−a, a] is zero.
Geometrical Interpretation of Integrals
Integrals are usually used for calculating the areas under the curve. Although there are formulas available, they are only available for standard shapes.
Often arbitrarily complex shapes are encountered, and it is not possible to develop and remember a formula for every shape. Thus, integrals help in generalizing the method for calculating the areas and the volume. Consider a function f(x), the objective is to calculate the area of the function.
The function is divided up into different parts in the shape of rectangles, these parts add up and form the area. The width of each rectangle is
As the number of rectangles increase, the width of these rectangles becomes very small and can be denoted by "dx". The area of each rectangle becomes f(x)dx. The total area is the sum of areas of all these small rectangles,
A(x) = Σ(x) = ∫ f(x) dx
This is explained in the image added below:
Solved Questions on Integrals
Question 1: Find the integral for the given function f(x) = 2sin(x) + 1.
Solution:
Given, f(x) = 2sin(x) + 1
∫ f(x) dx = ∫ (2 sin x + 1) dx
= ∫ 2 sin x dx + ∫1 dx
= - 2 cos x + x + C
Question 2: Find the integral for the given function f(x) = 2x2 + 3x.
Solution:
Given, f(x) = 2x2 + 3x
∫ f(x) dx = ∫ (2x2 + 3x) dx
= 2 ∫ x2 dx + 3 ∫ x dx
= 2 (x3/3) + 3x + C
Question 3: Find the integral for the given function f(x) = x3 + 3x2 + x + 1.
Solution:
Given, f(x) = x3 + 3x2 + x + 1
∫ f(x) dx = ∫ (x3 + 3x2 + x + 1)dx
= ∫x3 dx + 3∫x2 dx + ∫xdx + ∫1dx
= x4/4 + 3x3/3 + x2/2 + x + c
Question 4: Find the integral for the given function f(x) = sin(x) + 5cos(x).
Solution:
Given, f(x) = sin(x) + 5cos(x)
∫ f(x) dx = ∫ {sin x + 5 cos x}
= ∫ (sin x) dx + 5∫ cos x dx
= cos x + 5(-sin x) + c
= cos x - 5 sin x + c
Question 5: Find the integral for the given function f(x) = 5x-1 + 12.
Solution:
Given, f(x) = 5x-1 + 12
∫ f(x) dx = ∫ (5/x + 12) dx
= 5∫(1/x).dx + 12∫1.dx
= 5(ln x) + 12x + c
Question 6: Find the integral for the given function f(x) = (x + 1/x).
Solution:
Given,
∫ f(x) = ∫ (x + 1/x)dx
= ∫xdx + ∫(1/x)dx
= x2/2 + ln x + c
Integrals Practice Questions
Q1. Find the integral for the given function f(x) = (x + 1/x).
Q2. Find the integral for the given function f(x) = (2x3 + sin x).
Q3. Find the integral for the given function f(x) = (x2 + x + 1)/(x).
Q4. Find the integral for the given function f(x) = (t + 1)/(t2 - 1).