Integration Formulas: Basic, Trigonometric and Advanced

Various mathematical methods are used for calculations, including functions, differentiation, and integration. Integration is considered the reverse process of differentiation, also known as inverse differentiation. It involves finding a function given its derivative.

Integrals are significant in multiple fields, such as mathematics, science, and engineering. They are primarily used for calculating areas under simple curves, regions bounded by a curve and a line, and areas between two curves. Additionally, integrals have broad applications in mathematical disciplines.

Some key applications of integrals include determining the centre of gravity, calculating mass and momentum of inertia for vehicles, satellites, and towers, analysing the centre of mass, computing the velocity and trajectory of a satellite when placed in orbit, and measuring thrust. The blog covers more on the topic in detail:

Definition of Integration Formulas

Integration formulas are mathematical expressions that determine the integral of various functions, including algebraic expressions, trigonometric ratios, inverse trigonometric functions, and logarithmic and exponential functions.

Also known as integral calculus, these integral formulas help find the antiderivative of a function, essentially reversing the process of differentiation. One can recover the original function from its derivative by applying integration formulas.

Integration is broadly categorised into basic integration formulas, integration of trigonometric and inverse trigonometric functions, integration involving the product of functions, and advanced integration techniques. Fundamentally, integration serves as a method of combining parts to form a whole. Subsequent sections of the blog discuss integral calculus or integration formula list in detail:

Integration Basic Formulas

By applying fundamental theorems of integration, several generalised results are derived, commonly referred to as integration formulas in indefinite integration. These integral formulas help determine the integrality of different functions. Some essential integration basic formulas include:

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

Integration Formulas of Trigonometric Functions

Integration formulas of trigonometric functions are used to evaluate integrals involving trigonometric expressions. These functions are often simplified and rewritten in an integrable form before applying the formulas. Below is a list of integral formulas for trigonometric functions:

• ∫ cos x dx< = sin x + C
• ∫ sin x dx = -cos x + C
• ∫ sec² x 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

A set of integration formulas of inverse trigonometric functions is provided below, which are useful for solving integral problems involving these functions:

• ∫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

Below are some advanced integration formulas crucial in solving complex integral problems.

• ∫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

There are three different integration formulas, each with its own unique technique for finding integrals. These methods lead to standardised integration formulas that can be applied based on the structure of the function being integrated. The different formula of integration are as follows:

Application of Integrals

Application of integrals can be observed in various fields, such as:

• Estimating the area enclosed by curves.
• Solving problems related to average distance, velocity, and acceleration.
• Finding the average value of a function.
• Approximating the volume and surface area of solids.
• Determining the centre of mass and work.
• Estimating arc length.
• Calculating the kinetic energy of a moving object using improper integrals.

Usually, integration is divided into two types: definite integration and indefinite integration, which will be discussed below:

Definite Integration Formula

A definite integral has fixed upper and lower limits, resulting in a specific numerical value. The definite integration formula is expressed as:
∫_a^b g(x) dx = G(b) - G(a), where g(x) = G'(x).

Indefinite Integration Formula

An indefinite integral does not have set limits, producing a general function that includes an arbitrary constant C. The indefinite integration formula is expressed as:
∫ g'(x) = g(x) + C

Wrapping Up

Integration is a fundamental concept in calculus, the inverse differentiation process. It is crucial in solving mathematical problems across various fields, including mathematics, science, and engineering. Integration has widespread applications, from determining areas under curves and solving kinematic equations to calculating mass, momentum, and centre of gravity.
Integrals help evaluate complex mathematical expressions efficiently. Using different types of integration methods, such as integration by parts, substitution, and partial fractions, along with a range of essential formulas, they provide a structured approach to solving real-world and theoretical problems. Whether used for definite or indefinite integration, integrals help.

FAQs

Q1: What are integration formulas?

A1: Integration formulas are basic tools used to solve a wide range of integral problems. They help determine the integrals of algebraic expressions, trigonometric functions, inverse trigonometric functions, and logarithmic and exponential functions. 

Q2: What is the use of integration formulas?

A2: Integration formulas have varied uses in mathematics and real-world scenarios. They are used to determine the length of a curve, calculate the area under a curve, and approximate the values of functions.
Integrals also help analyse object motion by determining their paths. Additionally, they are essential for finding irregular shapes' surface area and volume. Integration formulas are also used to identify the centre of mass or centre of gravity in physical systems.

Q3: What is the integration formulas of uv?

A3: The integration formula of UV is given as: ∫ uv dx = u ∫ v dx – ∫ (u' ∫ v dx) dx.

Q4: What are the 5 basic integration formulas?

A4: The five basic integration formulas are as follows:

• d/dx {φ(x)} = f(x) ⇔ ∫f(x) dx = φ(x) + C
• ∫ xn dx = xⁿ⁺¹/n+1+ C, n ≠ -1
• ∫(1/x) dx = log_e|x| + C
• ∫e^x dx = e^x + C
• ∫a^x d^x = (a^x / log_e a) + C

Q5: What are the rules of integration?

A5: There are 5 rules of integration, as discussed below:

• Power Rule states that integrating a function with an exponent follows a specific formula.
• Sum Rule explains that the integral of a sum of functions is the sum of their individual integrals.
• Difference Rule is similar to the sum rule but applies to the difference of functions.
• Multiplication by a Constant state that a constant factor can be taken outside the integral.
• Product Rule applies to the integration of the product of two functions.

Q6: What is called integration?

A6: Integration is the summation of discrete data and is used to determine functions that describe quantities like area, displacement, and volume, which cannot be measured individually.

Q7: What is dx in integration?

A7: In integration, dx is the differential of the variable x and indicates that the function is being integrated with respect to x.