[Maths Class Notes] on Integration by Parts Rule Pdf for Exam

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Mathematics as a subject requires a good amount of practice and conceptual clarity. Students often find the subject of Math to be difficult when compared to other subjects. So, to solve all your problems we have presented and explained the topic of Math in the simplest language. 

Let’s start learning Math together and you no longer will have any fear about the subject. In this particular article, students will get to learn about the following concepts – 

What is Integration?

  • In Mathematics, when general operations like addition operations cannot be performed, we use Integration to add values on a large scale.

  • There are different types of 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 antidifferentiation or Integration.

[int_b^a f(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

 

Types of Integration Math or the Integration Techniques

Here’s a list of Integration Methods –

  1. Integration by Substitution

  2. Integration by Parts Rule

  3. Integration by Partial Fraction

  4. Integration of Some particular fraction

  5. Integration Using Trigonometric Identities

In this article we are going to discuss the Integration by Parts rule, Integration by Parts formula, Integration by Parts examples, and Integration by Parts examples and solutions.

 

Integration by Parts Rule

  • 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 the Integration by Parts rule.

  • Let us take an integrand function that is equal to u(x) v(x).

  • In Mathematics, Integration by parts basically uses the ILATE rule that helps to select the first function and second function in the Integration by Parts method.

  • Integration by Parts formula,

            [int u(x).v(x) dx = u(x) int v(x).dx – (u′(x) int v(x).dx).dx]

The Integration by Parts formula, 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 

From the Integration by Parts formula discussed above,

 

Ilate Rule

In Integration by Parts, we have learned when the product of two functions is given to us then we apply the required formula. The integral of the two functions is taken, by considering the left term as the first function and the second term as the second function. This method is called the Ilate rule. Suppose, we have to integrate xex, then we consider x as the first function and ex as the second function. So basically, the first function is chosen in such a way that the derivative of the function could be easily integrated. Usually, the preference order of this rule is based on some functions such as Inverse, Algebraic, Logarithm, Trigonometric, Exponent. This rule helps us to solve Integration by Parts examples using the Integration by Parts formula.

 

What is the ILATE rule?

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

  • I-Inverse

  • L-Logarithmic

  • Algebraic

  • T-Trigonometric

  • E-Exponential

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

 

Note

  • Integration by Parts rule is not applicable for functions such as [int sqrt{x sin x dx}].

  • We do not add any constant while finding the integral of the second function.

  • Usually, if any function is a power of x or a polynomial in x, then we take it as the first function. However, in cases where another function is an inverse trigonometric function or logarithmic function, then we take them as the first function.

 

Rules to be Followed for Solvin
g Integration by Parts Examples:

So we followed these steps:

  • Choose u and v functions

  • Differentiate u: u’

  • Integrate v: ∫v dx

  • Put u, u’ and put ∫v dx into the given formula: u∫v dx −∫u’ (∫v dx) dx

  • Simplify and solve the Integration by Parts examples

In simpler words, to help you remember, the following ∫u v dx becomes:

(u integral v) – integral of (derivative u, integral v)

 

Standard Integrals in Integration

[int x^n dx]

[frac{x^{n+1}}{n+1} + C] where n≠

-1.

[int sinx dx ]

[- cos x + C]

[int cosx dx ]

[sin x + C]

[int sec^2x dx ]

[tan x +C ]

[int cosec^2x dx ]

[- cot x + C ]

[int secx tanx dx ]

[sec x + C ]

[int cosecx cotx dx ]

[- cosec x + C ]

 

Let’s understand better by solving Integration by Parts examples and solutions.

 

Questions to be Solved

Question 1. What is ∫x cos(x) dx ?

 

Answer : We have x multiplied by cos(x), so Integration by Parts is a good choice.

First choose which functions for u and v:

So now, we have obtained it in the format ∫u v dx and we can proceed:

Differentiate u: u’ = x’ = 1

Integrate the v part : ∫v dx = ∫cos(x) dx = sin(x)

Now we can put it together and we get the answer:

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