Skip to main content

Integration by Parts | Formula, Derivation and Examples, Class 12 Math Notes Study Material Download Free PDF

Integration by Parts or Partial Integration, is a technique used in calculus to evaluate the integral of a product of two functions.

The formula for partial integration is given by:

∫ u dv = uv – ∫ v du

Where u and v are differentiable functions of x. This formula allows us to simplify the integral of a product by breaking it down into two simpler integrals.

Suppose we have two functions f(x) and g(x) and we have to find the integration of their product i.e., ∫ f(x).g(x) dx where it is not possible to further solve the product of this product f(x).g(x). 

This integration is achieved using the formula:

∫ f(x).g(x) dx = f(x) ∫ g(x) d(x) – ∫ [f'(x) {∫g(x) dx} dx] dx + c

Where f'(x) is the first differentiation of f(x).

This formula is read as:

Integration of the First Function multiplied by the Second Function is equal to (First Function) multiplied by (Integration of Second Function) – Integration of (Differentiation of First Function multiplied by Integration of Second Function).

Note: Integration by part concept was first proposed by the famous Brook Taylor in his book in 1715. He wrote that we can find the integration of the product of two functions whose differentiation formulas exist.

Integration By Parts Formula

Integration by parts formula is the formula that helps us to achieve the integration of the product of two or more functions. Suppose we have to integrate the product of two functions as

∫u.v dx 

where u and v are the functions of x, then this can be achieved using,

∫u.v dx = u ∫ v d(x) – ∫ [u’ {∫v dx} dx] dx + c

The order to choose the First function and the Second function is very important and the concept used in most of the cases to find the first function and the second function is ILATE concept.

Derivation of Integration By Parts Formula

Integration By Parts Formula is derived using the product rule of differentiation. Suppose we have two functions u and v and x then the derivative of their product is achieved using the formula,

d/dx (uv) = u (dv/dx) + v (du/dx)

Now to derive the integration by parts formula using the product rule of differentiation.

Rearranging the terms

u (dv/dx) = d/dx (uv) – v (du/dx)

Integrating both sides with respect to x,

∫ u (dv/dx) (dx) = ∫ d/dx (uv) dx – ∫ v (du/dx) dx

simplifying,

∫ u dv = uv – ∫ v du

Thus, the integration by parts formula is derived.

ILATE Rule

The ILATE rule tells us about how to choose the first function and the second function while solving the integration of the product of two functions. Suppose we have two functions of x u and v and we have to find the integration of their product then we choose the first function and the by ILATE rule.

The ILATE full form is discussed in the image below,

ILATE Rule Integration by Parts | Formula, Derivation and Examples, Class 12 Math Notes Study Material Download Free PDF

ILATE Rule of Partial Integration

The ILATE rules give us the hierarchy of taking the first function, i.e. if in the given product of the function, one function is a Logarithmic function and another function is a Trigonometric function. Now we take the Logarithmic function as the first function as it comes above in the hierarchy of the ILATE rule similarly, we choose the first and second functions accordingly.

NOTE: It is not always appropriate to use the ILATE rule sometimes other rules are also used to find the first function and the second function.

How to  Find Integration by Part?

Integration by part is used to find the integration of the product of two functions. We can achieve this using the steps discussed below,

Suppose we have to simplify ∫uv dx

Step 1: Choose the first and the second function according to the ILATE rule. Suppose we take u as the first function and v as the second function.

Step 2: Differentiate u(x) with respect to x that is, Evaluate du/dx.

Step 3: Integrate v(x) with respect to x that is, Evaluate ∫v dx.

Use the results obtained in Step 1 and Step 2 in the formula,

∫uv dx = u∫v dx − ∫((du/dx)∫v dx) dx

Step 4: Simplify the above formula to get the required integration.

Tabular Integration by Parts

Tabular integration, also known as the DI method of integration, is an alternative technique for evaluating integrals that involve repeated application of integration by parts.

This method is particularly useful when dealing with integrals where the product of functions can be integrated multiple times to reach a simple result.

Here’s how the tabular method works:

  1. Begin by writing down the functions involved in the integral in two columns: one for the function to differentiate (u) and another for the function to integrate (dv).
    • Start with the function to integrate (dv) on the left column and the function to differentiate (u) on the right column.
  2. Continue differentiating the function in the u column until you reach zero or a constant. At each step, integrate the function in the dv column until you reach a point where further integration is not necessary.
  3. Multiply the terms diagonally and alternate the signs (+ and -) for each term. Sum up these products to find the result of the integration.

Here’s an example to illustrate the tabular integration method:

Let’s evaluate the integral ∫x sin(x) dx.

  • Step 1: Create a table with two columns for u (function to differentiate) and dv (function to integrate):
udv
xsin(x)
  • Step 2: Differentiate the function in the u column and integrate the function in the dv column:
udv
x-cos(x)
1-sin(x)
0cos(x)
  • Step 3: Multiply the terms diagonally and alternate the signs:

(x)(-cos(x)) – (1)(-sin(x)) + (0)(cos(x)) = -x cos(x) + sin(x)

So, the result of the integral ∫x sin(x) dx is -xcos(x) + sin(x).

The tabular integration method is especially useful when dealing with integrals that involve functions that repeat upon differentiation or integration, allowing for a systematic and organized approach to finding the antiderivative.

Applications of Integration by Parts

Integration by Parts has various applications in integral calculus it is used to find the integration of the function where normal integration techniques fail. We can easily find the integration of inverse and logarithmic functions using the integration by parts concept.

We will find the Integration of the Logarithmic function and Arctan function using integration by part rule,

Integration of Logarithmic Function (log x)

Integration of Inverse Logarithmic Function (log x) is achieved using Integration by part formula. The integration is discussed below,

∫ logx.dx = ∫ logx.1.dx

Taking log x as the first function and 1 as the second function.

Using ∫u.v dx = u ∫ v d(x) – ∫ [u’ {∫v dx} dx] dx
⇒ ∫ logx.1.dx =  logx. ∫1.dx – ∫ ((logx)’.∫ 1.dx).dx
⇒ ∫ logx.1.dx = logx.x -∫ (1/x .x).dx
⇒ ∫ logx.1.dx = xlogx – ∫ 1.dx
⇒ ∫ logx.dx = x logx – x + C

Which is the required integration of logarithmic function.

Integration of Inverse Trigonometric Function (tan-1 x)

Integration of Inverse Trigonometric Functions (tan-1 x) is achieved using Integration by part formula. The integration is discussed below,

∫ tan-1x.dx = ∫tan-1x.1.dx

Taking tan-1 x as the first function and 1 as the second function.

Using ∫u.v dx = u ∫ v d(x) – ∫ [u’ {∫v dx} dx] dx
⇒ ∫tan-1x.1.dx = tan-1x.∫1.dx – ∫((tan-1x)’.∫ 1.dx).dx
⇒ ∫tan-1x.1.dx = tan-1x. x – ∫(1/(1 + x2).x).dx
⇒ ∫tan-1x.1.dx = x. tan-1x – ∫ 2x/(2(1 + x2)).dx
⇒ ∫tan-1x.dx = x. tan-1x – ½.log(1 + x2) + C

Which is the required integration of Inverse Trigonometric Function.

Integration by Parts Formulas

We can derive the integration of various functions using the integration by parts concept. Some of the important formulas derived using this technique are

  • ∫ ex(f(x) + f'(x)).dx = exf(x) + C
  • ∫√(x2 + a2).dx = ½ . x.√(x2 + a2)+ a2/2. log|x + √(x2 + a2)| + C
  • ∫√(x2 – a2).dx =½ . x.√(x2 – a2) – a2/2. log|x +√(x2 – a2) |  C
  • ∫√(a2 – x2).dx = ½ . x.√(a2 – x2) + a2/2. sin-1 x/a + C

Integration By Parts Examples

Example 1: Find ∫ ex x dx.

Solution:

Let I =  ∫ ex x dx

Choosing u and v using ILATE rule

u = x
v = ex

Differentiating u

u'(x) = d(u)/dx

⇒ u'(x) = d(x)/dx
⇒ u'(x) = 1

∫v dx = ∫ex dx = ex

Using the Integration by part formula,

⇒ I = ∫ ex x dx 
⇒ I = x ∫ex dx − ∫1 (∫ ex dx) dx
⇒ I = xex − ex + C
⇒ I = ex(x − 1) + C

Example 2: Calculate ∫ x sin x dx.

Solution:

Let I = ∫ x sin x dx

Choosing u and v using ILATE rule

u = x
v = sin x

Differentiating u

u'(x) = d(u)/dx
⇒ u'(x) = d(x)/dx
⇒ u'(x) = 1

Using the Integration by part formula,

⇒ I = ∫ x sin x dx 
⇒ I = x ∫sin x dx − ∫1 ∫(sin x dx) dx
⇒ I = − x cos x − ∫−cos x dx
⇒ I = − x cos x + sin x + C

Example 3: Find ∫ sin−1 x  dx.

Solution:

Let I=  ∫ sin−1 x  dx

⇒ I =  ∫ 1.sin−1 x  dx

Choosing u and v using ILATE rule

u = sin−1 x
v = 1

Differentiating u

u'(x) = d(u)/dx
⇒ u'(x) = d(sin−1 x )/dx 
⇒ u'(x) = 1/√(1 − x 2) 

Using the Integration by part formula,

⇒ I = ∫ sin−1 x dx 
⇒ I = sin−1 x ∫ 1 dx  − ∫ 1/√(1 − x 2)  ∫(1 dx)  dx
⇒ I  = x sin−1 x  − ∫( x/√(1 − x 2 ) )dx

Let, t = 1 − x 2 

Differentiating both sides

 dt = −2x dx
⇒ −dt/2 = x dx
⇒ I = ∫ sin−1 x dx =  x sin−1 x  − ∫−(1/2√t ) dt
⇒ I   = x sin−1 x  + 1/2∫t−1/2 dt
⇒ I  = x sin−1 x + t1/2 + C
⇒ I  = x sin−1 x + √(1 − x2 ) + C

FAQs on Integration by Parts

What is integration by parts?

Integration by parts is the technique for finding the integration of the product of the two functions where the normal techniques of integration fail. Integration by the part formula is the,

∫u.v dx = u ∫ v d(x) – ∫ [u’ {∫v dx} dx] dx + c

What is integration by parts formula?

For two functions f(x) and g(x) the integration by part formula is,

∫ f(x).g(x) dx = f(x) ∫ g(x) d(x) – ∫ [f'(x) {∫g(x) dx} dx] dx + c

where f'(x) is differentiation of f(x).

How to derive integration by parts formula?

Integration by part formula is derived using the product rule of differentiation.

Why do we use integration by parts formula?

Integration by part formula is used to find the integration of the function when the normal differentiation techniques fail. We can find the integration of inverse trigonometric functions, and logarithmic functions using Integration by part formula

What is the application of integration by parts?

Integration by part has various applications and the basic application of it is that it is used to find the integration of the function when the function is given as the product of the functions which can not be simplified further. For example ∫ f(x).g(x) dx is achieved using Integration by parts.

Labels:

Comments

Post a Comment

Popular posts from this blog

Symmetric & Skew Symmetric Matrix-Properties, Solved Examples, Class 12 Matrices Chapter Notes Study Material Download pdf

A symmetric matrix and skew-symmetric matrix both are square matrices. But the difference between them is, the symmetric matrix is equal to its transpose whereas skew-symmetric matrix is a matrix whose transpose is equal to its negative. If A is a symmetric matrix, then A = A T   and if A is a skew-symmetric matrix then A T  = – A. Table of Contents Symmetric Matrix Properties of Symmetric Matrix Skew Symmetric Matrix Properties of Skew Symmetric Matrix Determinant of Skew Symmetric Matrix Eigenvalue of Skew Symmetric Matrix Frequently Asked Questions-FAQs What is a symmetric matrix? How do you know if a matrix is symmetric? Give an Example of a Matrix Which is Symmetric but not Invertible. Is Symmetric Matrix Diagonalizable? What is skew-symmetric matrix? What is the difference between symmetric and skew-symmetric matrix? Symmetric Matrix To understand if a matrix is a symmetric matrix, it is very important to know about transpose of a matrix and how to find it. If we in...
Post a Comment
Read more

Onto Functions(Surjective Functions)-Definition, Graph, Properties, Solved Examples, FAQs

  Onto Function is one of the many types of functions defined based on the relationship between its domain and codomain. For any function to be onto, it needs to relate two sets with a very specific mapping between elements, meaning that each element of the codomain has at least one element in the domain as its pre-image. In simple words, for any function, if all the elements of the codomain are mapped to some element of the domain, then the function is said to be an onto function.  In this article, we will discuss the concept of onto or surjective function in detail including its definition, example, and many more. We will also discuss key differences between one one, onto and into functions as well. Table of Contents What is an Onto Function? Onto Function Definition Representation for Onto Function Examples of Onto Function Properties of Onto Function Composition of Onto Function Onto Function Graph Number of Onto Functions One to One and Onto Functions Onto and Into Functi...
Post a Comment
Read more

Matrices Types, Properties:Row, Column, Zero or Null, Singleton, Horizontal, Vertical, Square, Diagonal, Scalar, Unit or Identity Matrix, Equal Matrices, Triangular, Singular & Non-Singular Matrix, Symmetric & Skew Symmetric Matrices, Hermitian & Skew-Hermitian Matrices, Idempotent, Nilpotent, Periodic, Involutory Matrix

There are different   types of matrices,   and they are basically categorised on the basis of the value of their elements, their order, the number of rows and columns, etc. Now, using different conditions, the various matrix types are categorised below, along with their definition and examples. Table of Contents Matrix Types: Overview Types of Matrices: Explanations Row Matrix Column Matrix Zero or Null Matrix Singleton Matrix Horizontal Matrix Vertical Matrix Square Matrix Diagonal Matrix Scalar Matrix Unit Matrix or Identity Matrix Equal Matrices Triangular Matrix Singular Matrix and Non-Singular Matrix Symmetric and Skew Symmetric Matrices Hermitian and Skew-Hermitian Matrices Special Matrices (a) Idempotent Matrix: (b) Nilpotent Matrix: (c) Periodic Matrix: (d) Involutory Matrix: Frequently Asked Questions What do you mean by a symmetric matrix? What do you mean by identity matrix? What do you mean by Hermitian matrix? Matrix Types: Overview The different types of matrices...
Post a Comment
Read more