List Of Integration formulas

List Of Integration formulas

Integration formulas play a pivotal role in calculus, enabling mathematicians and scientists to find solutions to intricate problems related to areas, volumes, and rates of change. These formulas, derived from the fundamental theorem of calculus, provide a systematic way to evaluate integrals and solve real-world problems.

Integration Formulas

Integration formulas are the basic formulas that are used to solve various integral problems. They are used to find the integration of algebraic expressions, trigonometric ratios, inverse trigonometric functions, and logarithmic and exponential functions. These integration formulas are very useful to find the integration of various functions. 

Integration is the inverse process of differentiation, i.e. if d/dx (y) = z, then ∫zdx = y. Integration of any curve gives the area under the curve. We find the integration by two methods Indefinite Integration and Definite Integration. In indefinite integration, there is no limit of the integration whereas in definite integration there is a limit under which the function is integrated.

Let us learn about these integral formulas in detail in this article.

Integral Calculus

Integral calculus is a branch of calculus that deals with the theory and applications of integrals. The process of finding integrals is called integration. Integral calculus helps in finding the anti-derivatives of a function. The anti-derivatives are also called the integrals of a function. It is denoted by ∫f(x)dx. Integral calculus deals with the total value, such as lengths, areas, and volumes. The integral can be used to find approximate solutions to certain equations of given data. Integral calculus involves two types of integration: 

• Indefinite Integrals
• Definite Integrals

What are Integration Formulas?

The integration formulas have been broadly presented as the following sets of formulas. The formulas include basic integration formulas, integration of trigonometric ratios, inverse trigonometric functions, the product of functions, and some advanced sets of integration formulas. Basically, integration is a way of uniting the part to find a whole. It is the inverse operation of differentiation. Thus the basic integration formula is 

∫ f'(x) dx = f(x) + C. 

Using this, the following integration formulas are derived.

The various integral calculus formulas are

1. d/dx {φ(x)} = f(x)  <=> ∫f(x) dx = φ(x) + C
2. ∫ xⁿ dx =  + C, n ≠ -1
3. ∫(1/x) dx = log_e|x| + C
4. ∫e^x dx = e^x + C
5. ∫a^x dx = (a^x / log_ea) + C

More, integral formulas are discussed below in the article,

Note:

• d/dx [∫f(x) dx] = f(x)
• ∫k . f(x) dx = k ∫f(x) dx , where k is constant
• ∫{f(x) ± g(x)} dx = ∫f(x) dx  ± ∫g(x) dx

Basic Integration Formulas

Some of the basic formulas of integration which are used to solve integration problems are discussed below. They are derived by the fundamental theorem of integration.

• ∫ 1 dx = x + C
• ∫ xⁿ dx = x^{(n + 1)}/(n + 1)+ C
• ∫ 1/x dx = log |x| + C
• ∫ e^x dx = e^x + C
• ∫ a^x dx = a^x /log a+ C
• ∫ e^x [f(x) + f'(x)] dx = e^x f(x) + C   {where, f'(x) = d/dx[f(x)]}

Classification of Integral Formulas

Integral Formulas are classified into various categories based on the following function.

• Rational functions
• Irrational functions
• Hyperbolic functions
• Inverse hyperbolic functions
• Trigonometric functions
• Inverse trigonometric functions
• Exponential functions
• Logarithmic functions

Integration Formulas of Trigonometric Functions

Integration Formulas of Trigonometric functions are used to solve the integral equations involving Trigonometric functions. A list of integral formulas involving trigonometric and inverse trigonometric functions is given below,

• ∫ cos x dx = sin x + C
• ∫ sin x dx = -cos x + C
• ∫ sec²x dx = tan x + C
• ∫ cosec²x dx = -cot x + C
• ∫ sec x tan x dx = sec x + C
• ∫ cosec x cot x dx = -cosec x + C
• ∫ tan x dx = log |sec x| + C
• ∫ cot x dx = log |sin x| + C
• ∫ sec x dx = log |sec x + tan x| + C
• ∫ cosec x dx = log |cosec x – cot x| + C

Integration Formulas of Inverse Trigonometric Functions

Various Integration Formulas of Inverse Trigonometric Functions which are used to solve integral questions are given below,

• ∫1/√(1 – x²) dx = sin^-1x + C
• ∫ -1/√(1 – x²) dx = cos^-1x + C
• ∫1/(1 + x²) dx = tan^-1x + C
• ∫ -1/(1 + x²) dx = cot^-1x + C
• ∫ 1/x√(x² – 1) dx = sec^-1x + C
• ∫ -1/x√(x² – 1) dx = cosec^-1 x + C

Advanced Integration Formulas

Some other advanced integration formulas which are of high importance for solving integrals are discussed below,

• ∫1/(x² – a²) dx = 1/2a log|(x – a)(x + a| + C
• ∫ 1/(a² – x²) dx =1/2a log|(a + x)(a – x)| + C
• ∫1/(x² + a²) dx = 1/a tan^-1x/a + C
• ∫1/√(x² – a²)dx = log |x +√(x² – a²)| + C
• ∫ √(x² – a²) dx = x/2 √(x² – a²) -a²/2 log |x + √(x² – a²)| + C
• ∫1/√(a² – x²) dx = sin^-1 x/a + C
• ∫√(a² – x²) dx = x/2 √(a² – x²) dx + a²/2 sin^-1 x/a + C
• ∫1/√(x² + a² ) dx = log |x + √(x² + a²)| + C
• ∫ √(x² + a² ) dx = x/2 √(x² + a² )+ a²/2 log |x + √(x² + a²)| + C

Different Integration Formulas

Various types of integration methods are used to solve different types of integral questions. Each method is a standard result and can be considered a formula. Some of the important methods are discussed below in this article. Let’s check the three important integration methods.

• Integration by Parts Formula
• Integration by Substitution Formula
• Integration by Partial Fractions Formula

Integration by Parts Formula

Integration by Parts Formula is applied when the given function is easily described as the product of two functions. The integration by Parts formula used in mathematics is given below,

∫ f(x) g(x) dx = f(x) ∫g(x) dx – ∫ (∫f'(x) g(x) dx) dx + C

Example: Calculate ∫ xe^x dx

Solution:

∫ xe^x dx is of the form ∫ f(x) g(x) dx

let f(x) = x and g(x) = e^x

we know that, ∫ f(x) g(x) dx = f(x) ∫g(x) dx – ∫ (∫f'(x) g(x) dx) dx + C

∫ xe^x dx = x ∫e^x dx – ∫( 1 ∫e^x dx) dx+ c

              = xe^x – e^x + c

Integration by Substitution Formula

Integration by Substitution Formula is applied when a function is a function of another function. i.e. let I = ∫ f(x) dx, where x = g(t) such that dx/dt = g'(t), then dx = g'(t)dt

Now, I = ∫ f(x) dx = ∫ f(g(t)) g'(t) dt

Example: Evaluate ∫ (4x +3)³ dx

Solution:

Let u = (4x+3) ⇒ du = 4 dx

∫ (4x +3)³ dx 

= 1/4 ∫(u)³ du

= 1/4. u⁴ /5 

= u⁴ /20

= 4x +3)⁴/20

Integration by Partial Fractions Formula

Integration by Partial Fractions Formula is used when the integral of P(x)/Q(x) is required and P(x)/Q(x) is an improper fraction, such that the degree of P(x) is less than the (<) the  degree of Q(x), then the fraction P(x)/Q(x) is written as

P(x)/Q(x) = R(x) + P₁(x)/ Q(x)

where 
R(x) is a polynomial in x and 
P₁(x)/ Q(x) is a proper rational function. 

Now the integration of R(x) + P₁(x)/ Q(x) is easily calculated using the formulas discussed above.

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Application of Integral Formulas

Integral formulas are highly useful formulas in mathematics that are used for a variety of tasks. They are used for 

• Finding the length of the curve
• Finding the area under the curve
• Finding approximate values of the function
• Determining the path of an object and others
• To find the area under the curve
• To find the surface area and volume of irregular shapes
• To find the centre of mass or centre of gravity

These formulas are basically categorized into two categories,

• Definite Integration Formulas
• Indefinite Integration Formulas

Definite Integration Formulas

Definite integral formulas are used when the limit of the integration is given. In definite integration, the solution to the question is a constant value. Generally, the definite integration is solved as,

∫_a^b f(x) dx = F(b) – F(a)

Indefinite Integration Formulas

Indefinite Integration Formulas are used to solve the indefinite integration when the limit of integration is not given. In indefinite integration, we use the constant of the integration which is generally denoted by C

∫f(x) = F(x) + C

Solved Examples on Integral Formulas

Example 1: Evaluate

• ∫ x⁶ dx  
• ∫1/x⁴ dx  
• ∫³√x dx  
• ∫3^x dx  
• ∫4e^x dx  
• ∫(sin x/cos²x) dx  
• ∫(1/sin²x) dx 
• ∫[1/√(4 – x²)] dx  
• ∫[1/3√(x² – 9)] dx  
• ∫(1 /cos x tan x) dx 

Solution:

 (i)∫x⁶ dx 

= (x⁶⁺¹)/(6 + 1) + C       [∫xⁿ dx = {xⁿ⁺¹/(n+1)} + C    n ≠ -1]

= (x⁷/7) + C 

(ii) ∫1/x⁴ dx 

= ∫x^-4 dx                       [∫xⁿ dx = {xⁿ⁺¹/(n+1)} + C    n ≠ -1]

= (x^-4+1)/(-4 + 1) + C 

= -(x^-3/ 3) + C

= -(1/3x³) + C

(iii)  ∫³√x dx 

= ∫x^1/3 dx                     [∫xⁿ dx = {xⁿ⁺¹/(n+1)}+ C    n ≠ -1]

= (x ^(1/3)+1/((1/3)+ 1) + C 

= x^4/3 / (4/3) + C 

=  (3/4)(x^4/3) + C

(iv) ∫3^x dx 

= (3^x / log_e 3) + C        [ ∫a^x dx = (a^x / log_ea) + C]                                            

(v) ∫4e^x dx  

= 4∫e^x dx                     [∫k . f(x) dx = k f(x) dx , where k is constant]

= 4 e^x + C                   [∫e^x dx = e^x + C]

(vi) ∫(sin x/cos²x) dx 

= ∫[(sin x/cos x) .(1/cos x)] dx

= ∫tan x . sec x dx        [ ∫tan x .sec x dx = sec x + C ]       

= sec x + C

(vii) ∫(1/sin²x) dx 

= ∫cosec²x dx               [∫cosec²x dx  = -cot x + C ]

= -cot x  + C

(viii) ∫[1/√(4 – x²)] dx 

= ∫[1/√(2² – x²)] dx      [we know that, dx = sin^-1(x/a) + C]

= sin^-1(x/2) + C

(ix) ∫[1/{3√(x²  – 9)}] dx 

= ∫[1/{3√(x² – 3²)}] dx   [we know that, dx = (1/a)sec^-1(x/a) + C]

= (1/3)sec^-1(x/3) + C

(x) ∫(1 /cos x tan x) dx 

= ∫[cos x /(cos x sin x)] dx

= ∫(1/ sin x) dx

= ∫cosec x dx                [we know that, ∫cosec x dx =  log |cosec x – cot x| + C]

= log |cosec x – cot x| + C

Example 2: Evaluate ∫{e^9logex + e^8logex}/{e^6logex + e^5logex} dx

Solution:

 Since, e^alogex = x^a

∫{e^9logex + e^8logex}/{e^6logex + e^5logex} dx 

= ∫{x⁹ + x⁸}/{x⁶ + x⁵} dx

= ∫[x⁸(x + 1)]/[x⁵(x + 1)] dx

=∫ x⁸/x⁵ dx

= ∫x³ dx                    [we know that, ∫xⁿ dx = {xⁿ⁺¹/(n+1)} + C    n ≠ -1]

= (x⁴/4) + C 

Example 3: Evaluate ∫ sin x + cos x  dx 

Solution:

  ∫(sin x + cos x) dx 

= ∫sin x dx + ∫cos x dx             [we know that, ∫{f(x) ± g(x)} dx = ∫f(x) dx ± ∫g(x) dx]         

= -cos x + sin x + C                  [we know that, ∫sin x dx = -cos x + C, ∫cos x dx = sin x + C ]

Example 4: Evaluate ∫4^x+2 dx

Solution:

 ∫4^x+2 dx = ∫4^x. 4² dx 

= ∫16. 4^x dx              [ we known that∫k.f(x) dx = k∫f(x) dx , where k is constant]

= 16∫ 4^x dx               [∫a^x dx = (a^x / log_ea) + C]

= 16 (4^x/log 4) + C

Example 5: Evaluate ∫(x² + 3x + 1) dx

Solution:

 ∫(x² + 3x + 1) dx 

= ∫x² dx+ 3∫x dx + 1∫ x⁰dx         [We know that, ∫xⁿ dx = {xⁿ⁺¹/(n+1)}+ C  n ≠ -1]

= [x²⁺¹/2+1] + 3[[x¹⁺¹/1+1]] + [x⁰⁺¹/0+1] + C

= [x³/3] + 3[x²/2] + x + C

Example 6: Evaluate ∫[4/(1 + cos 2x)] dx 

Solution:

1 + cos 2x = 2cos²x 

∫[4/(1 + cos 2x)] dx 

= ∫[4/(2cos²x)] dx

= ∫(2/cos²x) dx

= ∫2 sec²xdx

= 2∫sec²x dx     [We know that, ∫sec²x dx = tan x + C ]

= 2 tan x + C 

Example 7: Evaluate ∫(3cos x – 4sin x + 5 sec²x) dx

Solution:

∫(3cos x – 4sin x + 5 sec²x) dx 

= ∫3cos x dx – ∫4sin x dx + ∫5sec²x dx   [∫k.f(x) dx = k ∫f(x) dx, where k is constant]

=  3∫cos x dx – 4∫sin x dx + 5∫sec²x dx

= 3sin x – 4(-cos x) + 5 tan x + C

=  3sin x + 4cos x + 5 tan x + C 

FAQs on Integration Formulas

Q1: What are all Integration Formulas?

Answer:

Integration formulas are the formulas which are used to solve various integration problems,

• ∫ 1 dx = x + C
• ∫ xⁿ dx = x^{(n + 1)}/(n + 1)+ C
• ∫ 1/x dx = log |x| + C
• ∫ e^x dx = e^x + C
• ∫ a^x dx = a^x /log a+ C
• ∫ e^x [f(x) + f'(x)] dx = e^x f(x) + C   {where, f'(x) = d/dx[f(x)]}

Q2: What is the integration formulas of uv?

Answer:

The integration formula of uv is,

∫uvdx = u∫vdx – ∫[d/dx(u) × ∫vdx] dx

Q3: What does integration in mathematics mean?

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Answer:

If the derivative of the function g(x) is f(x) then the integration of f(x) is g(x) i.e. ∫f(x)dx = g(x). Integration is represented by the symbol “∫”

Q4: How do we Integrate using Integration Formulas?

Answer:

Integration can be achieved using the formulas,

• Define a small part of an object in certain dimensions which by adding infinitely times makes the complete object.
• Using integration formulas over that small part along the varying dimensions get us the complete object.

Q5: What is the Integral Formula by Part?

Answer:

Integral formula by part is used to solve the integral where improper fraction is given.

Q6: What is the Use of Integration Formulas?

Answer:

Integration formulas are used to solve various integral problems. Various problems which we encounter in our daily life can be easily solved with the help of integration, such as finding center of mass of any object, finding the trajectory of missile, rockets, planes and others.

Integral formulae are key tools in calculus, used to calculate areas under curves, volumes of solids, and a variety of other things. Calculus’ basic theorem relates integrals with derivatives, making integral evaluation easier. The power rule, substitution rule, and integration by parts are all important formulae. Understanding and using these principles enables pupils to solve a wide range of mathematical problems.

Integral formulae for algebraic functions

Rules for integration:

Here are some common integration formulas for algebraic functions:

Power Rule:

• ∫ x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1.

Constant Multiple Rule:

• ∫ k * f(x) dx = k * ∫ f(x) dx, where k is a constant.

Sum/Difference Rule:

• ∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx.

Integration by Parts:

• ∫ u dv = u * v – ∫ v du, where u and v are differentiable functions.

These formulas are just a few examples of the wide range of algebraic functions that can be integrated. Integrating more complex functions may require using multiple techniques and approaches..

Integral formulae for Trigonometric functions:

Integration of hyperbolic function

Integration of rational functions

Integration of irrational function

Special integrals

Integration by substitution related formula

Integration formula by partial fractions

Integration by parts:

If f(x), g(x) are any two functions, then

∫f(x).g(x)dx = f(x)∫g(x)dx – ∫f'(x)∫g(x)dxdx

Proper choice of first and second function

• The first function is the function which comes first in the word ILATE
• If one of the two functions is not directly integrable, then take this function as the first function.
• If one of the function is not directly integrable, and there is no other function, then unity is taken as the second function.

Some Integral formulas using integration by parts

Integration formula for inverse trigonometric functions

Solved examples which including integration formulas

Solve some examples using the integration formulas mentioned earlier.

Example 1: ∫ (3x^2 + 2x + 1) dx

Using the Power Rule:

∫ (3x^2) dx = (3 * x^(2+1))/(2+1) + C

= (3/3) * x^3 + C

= x^3 + C.

Using the Power Rule:

∫ (2x) dx = (2 * x^(1+1))/(1+1) + C

= x^2 + C.

Using the Power Rule:

∫ (1) dx = x + C.

Putting it all together:

∫ (3x^2 + 2x + 1) dx = x^3 + x^2 + x + C.

Example 2: ∫ (e^x + 5sin(x)) dx

Using the Exponential Integral:

∫ e^x dx = e^x + C.

Using the Trigonometric Integral:

∫ sin(x) dx = -cos(x) + C.

Since the integral of a sum is the sum of integrals:

∫ (e^x + 5sin(x)) dx = (e^x + C) + 5*(-cos(x) + C)

= e^x – 5cos(x) + C.

Example 3: ∫ (x^3 + 2x^2 – 2x + 5) dx

Using the Power Rule:

∫ (x^3) dx = (x^(3+1))/(3+1) + C = (1/4) * x^4 + C.

Using the Power Rule:

∫ (2x^2) dx = (2 * x^(2+1))/(2+1) + C = (2/3) * x^3 + C.

Using the Power Rule:

∫ (-2x) dx = -2 * (x^(1+1))/(1+1) + C = -x^2 + C.

Using the Power Rule:

∫ (5) dx = 5 * x + C.

Putting it all together:

∫ (x^3 + 2x^2 – 2x + 5) dx = (1/4) * x^4 + (2/3) * x^3 – x^2 + 5x + C.

These are just a few examples to illustrate the use of integration formulas. Remember, integration can involve more complex functions and may require multiple steps or special techniques in some cases.

Frequently asked questions on Integration formulas

What is the integration formula?

A mathematical statement that allows us to compute the antiderivative of a given function is known as an integration formula. When the derivative of the original function is known, it aids in determining the original function. Integration allows the computation of areas, volumes, and other important quantities in calculus by employing various integration rules and procedures.

What is dx in integration?

In integration, the symbol dx represents the differential of the variable x. It is an infinitesimal change in the independent variable x. The process of integration involves finding the antiderivative of a function with respect to x, and the ‘dx’ in the integral notation indicates that the integration is being performed with respect to ‘x.’ The ‘dx’ notation is used to specify the variable of integration and is a fundamental part of the integral calculus notation. It allows us to find the area under a curve, calculate accumulated quantities, and solve a wide range of mathematical problems.

What is called integration?

Integration is a fundamental concept in calculus, and it refers to the process of finding the antiderivative of a function. It is the reverse operation of differentiation. When we integrate a function, we determine another function whose derivative is equal to the original function. Integration is used to calculate areas under curves, volumes of solids, and to solve various mathematical problems in physics, engineering, economics, and other fields. It plays a crucial role in understanding the behavior of functions and their cumulative effects.

Why is integration used?

Integration is a foundational technique in various fields, utilized for calculating areas under curves, volumes of shapes, and accumulated quantities. It’s pivotal in physics and engineering for determining work, energy, and other key variables. In probability and statistics, it’s applied to functions like probability densities. Integration also plays a critical role in solving differential equations, economic and financial modeling, signal processing, and optimization tasks, making it essential for both theoretical understanding and practical problem-solving across disciplines.

What are the rules of integration?

The rules of integration, also known as integration techniques or methods, are essential tools in calculus used to find antiderivatives and solve integrals. Improper Integrals: Used to evaluate integrals with infinite limits or unbounded functions. These rules, along with other specialized techniques, allow us to evaluate a wide range of integrals and solve diverse mathematical problems.

List of Integrals -formula concepts video