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Integration by Parts | Mathematics (Maths) Class 12 - JEE PDF Download

Definition

Integration by Parts | Mathematics (Maths) Class 12 - JEEdx where u & v are differentiable functions.

Note : While using integration by parts, choose u & v such that
(a) Integration by Parts | Mathematics (Maths) Class 12 - JEE dx v is simple &  

(b) Integration by Parts | Mathematics (Maths) Class 12 - JEE dx is simple to integrate.

This is generally obtained, by keeping the order of u & v as per the order of the letter in ILATE, where

I – Inverse function
L – Logarithmic function
A – Algebraic function
T – Trigonometric function
E – Exponential function

Remember This:
Integration by Parts | Mathematics (Maths) Class 12 - JEE

Proof:

Integrating by parts taking sin bx as the second function,

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Again integrating by parts taking cos bx as the second function, we get
Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Transposing the term -a2/b2 I to the left hand side, we get  Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE


Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE


Solved Examples

Ex.1 Integrate xlog x

Sol.

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Ex.2 Evaluate Integration by Parts | Mathematics (Maths) Class 12 - JEEdx

Sol.

Put sec–1 x = t so that  Integration by Parts | Mathematics (Maths) Class 12 - JEE

Then the given integral  =  Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

= t (log t – log e) + c  = sec–1 x (log (sec–1 x) – 1) + c   Integration by Parts | Mathematics (Maths) Class 12 - JEE

 

Ex.3 Evaluate Integration by Parts | Mathematics (Maths) Class 12 - JEEdx.

Sol.

Put x = cos θ so that dx = - sin θ dθ. the given integral 

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE


Ex.4 Evaluate Integration by Parts | Mathematics (Maths) Class 12 - JEE

Sol.

We have  Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE [ x3 = x(x2 + 1) - x]

integrating by parts taking x2 as the second function 

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Ex.5 Evaluate Integration by Parts | Mathematics (Maths) Class 12 - JEEdx.


Sol.

  Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE (put, 2x + 2 = 3 tanθ ⇒ 2 dx = 3 sec2θ dθ )

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

 

Ex.6 If   Integration by Parts | Mathematics (Maths) Class 12 - JEE

Sol.

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE


Ex.7 Evaluate Integration by Parts | Mathematics (Maths) Class 12 - JEE


Sol.

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Ex.8 Evaluate  Integration by Parts | Mathematics (Maths) Class 12 - JEE

Sol.

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE x where f(x) = tan x = ex f(x) + c = ex tanx + c

 

Ex.9 Evaluate  Integration by Parts | Mathematics (Maths) Class 12 - JEE

Sol.

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

using, previous example Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

Integration by Parts | Mathematics (Maths) Class 12 - JEE

 

 

The document Integration by Parts | Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.
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FAQs on Integration by Parts - Mathematics (Maths) Class 12 - JEE

1. What is integration by parts?
Ans. Integration by parts is a technique used in calculus to evaluate the integral of a product of two functions. It is based on the product rule for differentiation and allows us to transform an integral into a different form that may be easier to solve.
2. How does integration by parts work?
Ans. Integration by parts works by applying the formula: ∫ u dv = uv - ∫ v du where u and v are functions of x, and du and dv are their respective differentials. By choosing appropriate u and dv, we can simplify the integral on the right-hand side and make it easier to evaluate.
3. When should integration by parts be used?
Ans. Integration by parts should be used when the integral involves a product of two functions and other integration techniques like substitution or simplification are not applicable. It is particularly useful when one function differentiates to zero or when one function integrates to a simpler form.
4. Can integration by parts be used multiple times?
Ans. Yes, integration by parts can be used multiple times, especially when the resulting integral still involves a product of functions. Each iteration involves choosing new u and dv and applying the integration by parts formula again. However, it is important to note that each iteration may not always lead to a simpler integral.
5. What are some common mistakes to avoid when using integration by parts?
Ans. Some common mistakes to avoid when using integration by parts include incorrect choice of u and dv, not differentiating or integrating correctly, and not simplifying the resulting integral after applying the formula. It is also important to be mindful of the signs and constants while applying the integration by parts formula.
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