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In mathematics, an integral is the general form of a mathematical operation that allows one to calculate, for a given function, a quantity that is closely related to the function itself. The most common integrals are those that arise from the four fundamental operations of arithmetic: addition, subtraction, multiplication, and division. Here Knowledge glow provide guide on Integral Formulas & list of formulas.

## What is an Integral Formula?

An **integral formula** is a mathematical formula used to calculate the value of a definite integral. There are many different types of integral formulas, each of which can be used to calculate a specific type of definite integral. The most common type of integral formula is the power rule, which can be used to calculate the value of any polynomial function. Other common types of **integral formulas** include the trapezoidal rule, the Simpson’s rule, and the midpoint rule. Each of these formulas can be used to calculate a specific type of definite integral, and each has its own advantages and disadvantages.

## The Different Types of Integral Formulas

**Integral formulas **are mathematical equations that allow you to calculate the area or volume of an object. There are three main types of integral formulas: surface area, arc length, and volume.**Surface area formulas** allow you to calculate the total surface area of an object. This is important for things like painting or tiling a room, where you need to know the exact amount of material you’ll need.**Arc length formulas** allow you to calculate the length of a curve. This is useful for things like calculating the distance around a race track, or the length of a river.**Volume formulas** allow you to calculate the amount of space an object takes up. This is important for things like estimating how much water you need to fill a swimming pool, or how many boxes you’ll need to move your belongings into a new home.

### The list of basic integral formulas are

- ∫ 1 dx = x + C
- ∫ a dx = ax+ C
- ∫ x
^{n }dx = ((x^{n+1})/(n+1))+C ; n≠1 - ∫ sin x dx = – cos x + C
- ∫ cos x dx = sin x + C
- ∫ sec
^{2}x dx = tan x + C - ∫ csc
^{2}x dx = -cot x + C - ∫ sec x (tan x) dx = sec x + C
- ∫ csc x ( cot x) dx = – csc x + C
- ∫ (1/x) dx = ln |x| + C
- ∫ e
^{x }dx = e^{x}+ C - ∫ a
^{x }dx = (a^{x}/ln a) + C ; a>0, a≠1

**Some other important integration formulas are:**

## Integration Formulas of Trigonometric functions

Integrals are a way to calculate the integral calculus of **trigonometric functions**. Here are a few formulas that we remember to quickly find the answer. There is really just one way to find what the integral of xsin(x) dxdx is: It’s just plain old sin(x). However, there are tons of resources on Google and YouTube to help you figure out integrals of all shapes and sizes.

- ∫ sin x dx = -cos x + C
- ∫ cos x dx = sin x + C
- ∫ sec
^{2}x dx = tan x + C - ∫ cosec
^{2}x 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

*Also Check : Important Maths Formulas for Class 8*

## Integration Formulas of Inverse Trigonometric Functions

Here we are showing the integral formulas that lead to give the result in the form of inverse trigonometric functions.

- ∫1/√(1 – x
^{2}) dx = sin^{-1}x + C - ∫ 1/√(1 – x
^{2}) dx = -cos^{-1}x + C - ∫1/(1 + x
^{2}) dx = tan^{-1}x + C - ∫ 1/(1 + x
^{2}) dx = -cot^{-1}x + C - ∫ 1/x√(x
^{2}– 1) dx = sec^{-1}x + C - ∫ 1/x√(x
^{2}– 1) dx = -cosec^{-1 }x + C

## Advanced Integration Formulas

Integral formulas are important when you’re solving problems of integration. To help you with these, we are provides some Advanced Integration Formulas.

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

## Pros and Cons of Integral Formulas

Integral formulas are mathematical expressions used to calculate the area under a curve. They are essential for solving many problems in calculus, but can be difficult to understand and use. We’ll explore the pros and cons of integral formulas to help you decide whether they are right for you.

On the plus side, integral formulas are very powerful tools. They can be used to solve complex problems that would be otherwise impossible to solve. Additionally, once you understand how to use them, they can be relatively easy to apply.

However, there are some downsides to integral formulas as well. They can be difficult to learn and understand, which can make them frustrating to use. Additionally, they can be computationally intensive, meaning they may take longer to calculate than other methods.

ultimately, whether or not integral formulas are right for you depends on your needs and abilities. If you need to solve complex calculus problems, then they may be worth the effort required to learn them. However, if you only need to solve simple problems, there may be easier methods available that would better suit your needs.

## Alternatives to Integral Formulas

When it comes to finding the area under a curve, there are a few different methods that can be used. One of the most common is using integral formulas, but there are some alternatives that can be used as well.

One popular alternative is the trapezoidal rule. This method involves dividing the area under the curve into a series of trapezoids and then calculating the area of each one. This can be a very accurate method, but it can be time-consuming if there are a lot of trapezoids involved.

Another alternative is Simpson’s Rule. This method uses a similar approach to the **trapezoidal rule**, but instead of trapezoids, it uses parabolas. This can make the calculations easier and faster, but it can also introduce error if the curve isn’t well-behaved.

Ultimately, the best method to use will depend on the situation. If accuracy is paramount, then using integral formulas is probably the best way to go. But if speed is more important, then one of the alternatives may be a better choice.

## Conclusion

Integral formulas are a type of mathematical formula that allows for the determination of integrals. These formulas can be used to solve problems in calculus and other areas of mathematics. While there is no one definitive list of integral formulas, there are some that are more commonly used than others. This article has provided an overview of some of the most popular formulas, as well as their applications. With this information in hand, you should be able to better understand how integral formulas can be used to solve problems in mathematics.