Methods of Integration

What is Integration?

  • In Mathematics, when we cannot perform general addition operations, we use integration to add values on a large scale.

  • There are various methods in mathematics to integrate functions.

  •  Integration and differentiation are also a pair of inverse functions similar to  addition- subtraction, and multiplication-division. 

  • The process of finding functions whose derivative is given is named anti-differentiation or integration. (image will be updated soon).

baf(x)dx = value of the anti-derivative at upper limit b – the value of the same anti-derivative at lower limit a.


Here’s What Integration is!

If \[\frac{d}{{dx}}(F(x)) = f(x),\] then


 \[\int {f(x)\,dx = F(x) + c} \]


The function F(x) is called anti-derivative or integral or primitive of the given function f(x) and c is known as the constant of integration or the arbitrary constant.


The function f(x) is called the integrand and f(x)dx is known as the element of integration.

 

Points to Remember:

Since the integral of a function isn’t definite, therefore it is generally referred to as indefinite integral.

We can never find the integral of a function at a point; we always find the integral of a given function in an interval.

Integral of a function is not unique; integrals of a function differ by numbers.

 

Types of Integration Maths or the Integration Techniques-

Here’s a list of Integration Methods –

1.Integration by Substitution

2. Integration by Parts

3.Integration by Partial Fraction

4.Integration of Some particular fraction

5.Integration Using Trigonometric Identities

For better understanding here’s what each method is!

1. Integration by Substitution -

  • We can find the integration by introducing a new independent variable when it is difficult to find the integration of a function. 

  • By changing the independent variable x to t, in a given form of integral function say \[\left( {\int {f(x)} } \right)\], we can transform the integral.

Let’s substitute the value of independent x = g(t) in the integral function ∫f(x), 

We get, dx / dt = g’(t)

Or, dx = g’(t) • dt

Thus, from the above substitution ,we get,

\[I = \int {f(x).dx = f(g(t).g'(t)).dt} \]

 

2. Integration by Parts –

  • If the integrand function can be represented as a multiple of two or more functions, the integration of any given function can be done by using Integration by Parts method.

  • Let us take an integrand function which is equal to f(x)g(x).

  • In mathematics, Integration by part uses the ILATE rule for selecting the first and second functions in this method.

  • In mathematics, here’s how integration by parts is represented.

∫f(x).g(x).dx = f(x).∫g(x).dx – ∫(f′(x).∫g(x).dx).dx

Which can be further written as integral of the product of any two functions = (First function × Integral of the second function) – Integral of [ (differentiation of the first function) × Integral of the second function]


What is the LIATE Rule?

LIATE is a rule which helps to decide which term should you differentiate first and which term should you integrate first.

  • L- Logarithm

  • I -Inverse

  • Algebraic

  • T-Trigonometric

  • E-Exponential

The term which is closer to L is differentiated first and the term which is closer to E is integrated first.

 

3. Integration Using Trigonometric Identities –

  • Trigonometric identities are used to simplify any integral function which consists of trigonometric functions.

  • It simplifies the integral function so that it can be easily integrated.

  • There are many trigonometric identities, a few are listed below!


Sin2 x = (1 - Cos 2x) / 2

 

Cos2x = (1 + Cos 2x) / 2

 

4. Integration of Some Particular Function -

  • Many other standard integrals that can be integrated using some important integration formulas.

  • Here are the six important formulas listed below - 

  • ∫ dx/ (x2 – a2) = ½  a log | (x – a) / (x + a) | + c

  • ∫ dx/ (a2 – x2) = ½ a log | (a + x) / (a – x) | + c

  • ∫ dx / (x2 + a2) = 1/a tan–1 (x/a) + c

  • ∫ dx /√ (x2 – a2) = log| x+√(x2 – a2) | + c

  • ∫ dx /√ (a2 – x2) = sin–1 (xa) + c

  • ∫ dx /√ (x2 + a2) = log | x + √(x2 + a2) | + c

Where, c = constant

 

5. Integration by Partial Fraction -

  • The partial fraction method is the last method of integration class 12.

  • In mathematics, rational numbers can be expressed in the form of \[\frac{p}{q}\] where p and q are integers and where the value of the denominator q is not equal to zero.

  • The ratio of two polynomials is known as a rational fraction and it can be expressed in the form of \[\frac{{p(x)}}{{q(x)}}\] , where the value of p(x) should not be equal to zero.

  • The two forms of partial fraction have been described below-

            Proper Partial function 

            Improper Partial function

  • What is the proper partial function?

When the degree of the denominator is more than the degree of the numerator, the function is known as a proper partial function.

  • What is improper partial function?

When the degree of the denominator is less than the degree of the numerator then the fraction is known as improper partial function. Thus, the fraction can be simplified into parts and can be integrated easily.

 

Questions to be Solved on Methods of Integration-

Question 1) Find the integration of the question using methods of integration.

\[\int {{{\tan }^4}\theta \,d\theta } \]

Solution) From the types of integration maths we know that, 

The above-given question can be solved using Trigonometric identities,

Let \[I = \int {{{\tan }^4}\theta \,d\theta }  = \int {({{\sec }^2}\theta  - 1){{\tan }^2}\theta \,d\theta } \]

\[ = \int {{{\sec }^2}\theta \,{{\tan }^2}\theta \,d\theta  - } \int {{{\tan }^4}\theta \,d\theta } \]

= By substitution, u = tan, the first part can be easily solved, 

\[ = \frac{1}{3}{\tan ^3}\theta \]

= The second integral can be solved by \[ = \int {({{\sec }^2}\theta  - 1)\,d\theta  = \tan \theta  - \theta  + C} \].

=Hence, \[I = \frac{1}{3}{\tan ^3}\theta  - \tan \theta  - \theta  + C\]

FAQ (Frequently Asked Questions)

Question 1) How many Methods of Integration Class 12 are There?

Answer) There are five methods of integration class 12 that are generally used-

Integration by Substitution

Integration by Parts

Integration by Partial Fraction

Integration of Some particular fraction

Integration Using Trigonometric Identities

Question 2) What is the LIATE Rule in Integration Techniques?

Answer) LIATE rule in integration technique is a rule which helps to decide which term should you differentiate first and which term should you integrate first.

L- Logarithm

I -Inverse

A- Algebraic

T-Trigonometric

E-Exponential

The term which is closer to L is differentiated first and the term which is closer to E is integrated first in integration methods.

Question 3) What is Integration?

Answer) The process of finding functions whose derivative is given is named anti-differentiation or integration. There are five integration methods.