Integral Formulas: List of Formulas & Methods of Integration

Integral Formulas are mathematical formulas used to solve various expressions and problems related to Integrals. Integrals are the anti-derivatives of a function calculated using Integration. They are the fundamental objects in Calculus and represent the area of a region under a curve. They are also known as antiderivative and primitive of a function. Integration is defined as the process of calculating an integral or anti-derivative. 

Integral Formulas are used for the integration of the following functions:

• Trigonometric Functions
• Inverse Trigonometric Functions
• Hyperbolic Functions
• Inverse Hyperbolic Functions
• Logarithmic Functions
• Exponential Functions
• Rational Functions
• Irrational Functions
• Gaussian Functions

Read More: NCERT Solutions For Class 12 Mathematics Integrals

Key Terms: Integrals, Integral Formulas, Integral Calculus, Integration, Differential Calculus, Definite Integral, Indefinite Integral

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What is Integral Calculus?

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Integral Calculus is a branch of Calculus that deals with the theory, properties, and applications of integrals. Integrals are the values of the function found through the process of Integration. 

• Integration is used to find the antiderivatives or integrals of a function.
• It is also known as the inverse of differentiation.
• f’ is called the derivative of the function f in differential calculus.
• f is called the anti-derivative or primitive of function f’ in integral calculus. 
• Integration is simply the process of getting f(x) from f'(x).
• Thus, integration is the process of finding anti-derivatives of a function.

Integral Calculus deals with two types of integration as follows: 

• Indefinite Integrals
• Definite Integrals

Introduction to Integrals Detailed Video Explanation

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What is Integration?

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Integration is defined as the process of finding the area of the region under the curve. It is a process of finding the antiderivative of a function.

• Integration is the inverse of differentiation. 
• It is used to define and calculate the area of the region bounded by the graph of functions.
• The principles of integration were formulated and proposed by Leibniz.
• There are two types of Integrals in Integration namely Definite Integral and Indefinite Integral.
• Indefinite Integrals do not have a pre-existing value of limits which means they cannot be defined using the upper and lower limits.
• Definite Integrals have a pre-existing value of limits which means they have upper and lower limits.

Definite Integration Formula

Definite Integration is the integration of definite integrals that have a pre-existing value of limits which makes the final value of the integral definite.

Indefinite Integration Formula

Indefinite Integration is the integration of indefinite integrals that do not have a pre-existing value of limits. It makes the final value of the integral indefinite. In indefinite integral, C is the integration constant.

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Integral Formulas

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Integral Formulas are a set of mathematical formulas used in the process of Integration. Integration is a method of integrating a part to find a whole. It is simply the inverse operation of Differentiation. The basic Integration formula is given as follows: 

This formula is further used to derive other important integral formulas.

Integral Formulas are classified into various types based on the following functions: 

• Trigonometric Functions
• Inverse Trigonometric Functions
• Exponential Functions
• Logarithmic Functions
• Rational Functions
• Irrational Functions
• Hyperbolic Functions
• Inverse Hyperbolic Functions
• Gaussian Functions

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List of Integral Formulas

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Here are standard integral formulas used in integral calculus:

• ∫ xⁿ dx=xⁿ⁺¹ /n+1+C, n ≠ -1
• ∫ dx =x+C
• ∫ cosxdx = sinx + C
• ∫ sinx dx = -cosx + C
• ∫ sec²x dx = tanx + C
• ∫ cosec²x dx = -cotx + C
• ∫ sec²x dx = tanx+C
• ∫ secx tanxdx = secx + C
• ∫ cscx cotx dx = -cscx + C
• ∫1/(√(1-x²)) = sin^-1 x + C
• ∫-1/(√(1-x²)) = cos^-1 x + C
• ∫1/(1+x²)= tan^-1 x + C
• ∫-1/(1+x²)= cot^-1 x + C
• ∫ e^xdx = e^x + C
• ∫dx/x = ln|x| + C
• ∫ a^x dx = a^x/ln a + C
• ∫1/(x√(x² -1)) = sec^-1 x + C
• ∫-1/(x√(x² -1)) = cosec^-1 x + C

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Basic Integration Formulas

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Using the fundamental theorem of integration, some basic formulas of integration are derived which are as follows: 

1. ∫ xⁿ. dx = x ^{(n + 1)}/ (n + 1) + C
2. ∫ 1.dx = x + C
3. ∫1/x. dx = log|x| + C
4. ∫ e^x. dx = e^x + C
5. ∫ a^x. dx = a^x /loga+ C
6. ∫ e^x[f(x) + f'(x)]. dx = e^x. f(x) + C

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Integration of Some Particular Functions

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Following are the formulas for the Integration of Some Particular Functions:

1. ∫ dx/ (x² – a²) is equal to 12 a log | (x – a) / (x + a) | + c
2. ∫ dx/ (a² – x²) is equal to1 2 a log | (a + x) / (a – x) | + c
3. ∫ dx / (x² + a²) is equal to1a tan^–1 (x/a) + c
4. ∫ dx /√ (x² – a²) is equal to log| x+√ (x² – a²) | + c
5. ∫ dx /√ (a² – x²) is equal to sin^–1 (xa) + c
6. ∫ dx /√ (x² + a²) is equal to log | x + √ (x² + a²) | + c

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Integration Formulas of Trigonometric Functions

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Integration Formulas of Trigonometric functions are used to solve the expression and equation involving Trigonometric functions. Trigonometric Functions are simplified and written as functions that are integrable.

Here are the Integration Formulas of Trigonometric Functions: 

1. ∫ cosx.dx = sinx + C
2. ∫ sinx.dx = -cosx + C
3. ∫ cosec²x.dx = -cotx + C
4. ∫ sec²x.dx = tanx + C
5. ∫ cosecx.cotx.dx = -cosecx + C
6. ∫ secx.tanx.dx = secx + C
7. ∫ tanx.dx =log|secx| + C
8. ∫ cotx.dx = log|sinx| + C
9. ∫ cosecx.dx = log|cosecx – cotx| + C
10. ∫ secx.dx = log|secx + tanx| + C

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Integration Formulas of Inverse Trigonometric Functions

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Integral Formulas that are used to give the output in the form of inverse trigonometric functions are as follows: 

1. ∫1/√ (1 – x²). dx = sin^-1x + C
2. ∫ /1(1 – x²). dx = -cos^-1x + C
3. ∫1/ (1 + x²). dx = tan^-1x + C
4. ∫ 1/ (1 +x²). dx = -cot^-1x + C
5. ∫ 1/x√ (x² – 1). dx = -cosec^-1 x + C
6. ∫ 1/x√ (x² – 1). dx = sec^-1x + C

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Advanced Integration Formulas

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Given below are some advanced integral formulas that are often used while solving problems in integration.

1. ∫ 1/ (a² – x²). dx =1/2a.log| (a + x) (a – x) | + C
2. ∫1/ (x² – a²). dx = 1/2a.log| (x – a) (x + a) | + C
3. ∫1/ (x² + a²). dx = 1/a.tan^-1x/a + C
4. ∫1/√ (x² – a²) dx = log|x| +√ (x² – a²) | + C
5. ∫1/√ (a² – x²). dx = sin^-1 x/a + C
6. ∫ √ (x² – a²). dx =1/2. x.√ (x² – a²)-a²/2 log|x| + √ (x² – a²) | + C
7. ∫√ (a² – x²). dx = 1/2. x.√ (a² – x²). dx + a²/2.sin^-1 x/a + C
8. ∫1/√ (x² + a²). dx = log|x| + √ (x² + a²) | + C
9. ∫ √ (x² + a²). dx =1/2. x.√ (x² + a²) + a²/2. log|x| + √ (x² + a²) | + C

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Methods of Integration

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Integrals of a function can be found using the three main methods of integration which are as follows: 

• Integration by Substitution Method
• Integration by Parts
• Integration by Partial Fractions

Integration by Substitution

If a given function is a function of another function, integration by substitution method is used to find the integral. If I = ∫ f(x) dx, where x = g(t) such that dx/dt = g'(t), then dx = g'(t) is written. It can be written as I = ∫ f(x) dx = ∫ f(g(t)) g'(t) dt.

In Integration by Substitution, if u is a function of x, then, u' = du/dx.

Where u = g(x).

Integration by Parts

Integration by Parts Method is used to find the integral of the product of two different types of functions. Integration By Parts Formula is given as follows: 

ILATE Rule

ILATE Rule is used in integration by parts for the selection process of the first function and the second function.

• I means Inverse Trigonometric Functions
• L means Logarithmic Functions
• A means Algebraic Functions
• T means Trigonometric Functions
• E means Exponential Functions

The term that is closer to the letter I is differentiated first and the term which is closer to the letter E is integrated first.

Integration by Partial Fraction

Integration by Partial Fractions is used to find the integral of an improper fraction like P(x)/Q(x), in which the degree of P(x) < Q(x). In this method, the fraction is split using partial fraction decomposition as P(x)/Q(x) = T(x) + P11 (x)/ Q(x), in which T(x) is a polynomial in x and P₁ (x)/ Q(x) is a proper rational function.

Assuming that A, B, and C are real numbers, there are the following types of simpler partial fractions associated with various types of rational functions.

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Integral Formulas Solved Examples

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Given below are some examples using Integral Formulas: 

Example 1: What is the value of ∫ cosec x(cosec x - cot x) dx?

Solution: Given that, ∫cosec x(cosec x - cot x) dx

= ∫ cosec²x dx - ∫ cot x cosec x dx

Using the trigonometric integration formulas, we get

= -cot x - (-cosec x)

= -cot x + cosec x = cosec x - cot x + C

Therefore, ∫ cosec x (cosec x - cot x) dx = cosec x - cot x + C

Example 2: What is the value of ∫(5 + 4cos x)/sin²x dx?

Solution: Given that, ∫(5 + 4cos x)/sin²x dx

= ∫5/sin²x. dx +∫4cos x/sin²x dx

= ∫5cosec²x dx + ∫4cot x cosec x dx

= 5∫cosec²x dx + 4∫cot x cosec x dx

According to the trigonometric integration formula, 

= 5(-cot x) + 4(-cosec x) = -5cot x - 4cosec x + C

Thus, ∫(5 + 4cos x)/sin²x dx = -5cot x - 4cosec x + C.

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Things to Remember

• Integral Formulas are a set of basic and complex formulas used in Integration. 
• Integrals are defined as the values of the function found by the process of integration. 
• Integrals are also referred to as anti-derivative of a function.
• Integral Formulas include basic integration formulas, integration of trigonometric and inverse trigonometric functions, etc.
• Definite Integrals and Indefinite Integrals are the two types of integrals in integration. 
• Integration is performed using integration by substitution method, integration by parts, and integration by partial fractions.

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Previous Year Questions 

1. For a>0, let the curves C_1​:y² = ax and C₂​:x² = ay intersect at origin O... (JEE Main – 2020)
2. The area of the region A=[(x,y) : 0≤y≤x∣x∣+1 and −1≤x≤1] in s units is... (JEE Main – 2019)
3. If \(∫xlog\,(1 + {1 \over x})\)= dx = f(x)log(x+1) + g(x)x² + Lx+C then... (BITSAT – 2017)
4. The area (in s units) of the region {(x,y) : x≥0, x + y≤3, x²≤4y and... (JEE Main – 2017)
5. If \({x^2 + 5} \over {(x^2 + 1) (x-2)} \)= \(A \over {x-2} \) + \(Bx + C \over {x^2 + 1}\), then A+B+C... (TS EAMCET - 2017)
6. You are given a curve, y=ln(x+e). What will be the area enclosed between... (JKCET - 2017)
7. The integral ∫cos (logx) dx is equal to… (JEE Main - 2019)
8. The area (in s units) bounded by the curves y... (COMEDK UGET - 2013)
9. Let the straight line x=b divide the area enclosed by... (AMUEEE - 2014)
10. The number of integral terms in the expansion…