Integration by Parts

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

## Definition

dx where u & v are differentiable functions.

Note : While using integration by parts, choose u & v such that
(a)  dx v is simple &

(b)  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:

Proof:

Integrating by parts taking sin bx as the second function,

Again integrating by parts taking cos bx as the second function, we get

Transposing the term -a2/b2 I to the left hand side, we get

## Solved Examples

Ex.1 Integrate xlog x

Sol.

Ex.2 Evaluate dx

Sol.

Put sec–1 x = t so that

Then the given integral  =

= t (log t – log e) + c  = sec–1 x (log (sec–1 x) – 1) + c

Ex.3 Evaluate dx.

Sol.

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

Ex.4 Evaluate

Sol.

We have

[ x3 = x(x2 + 1) - x]

integrating by parts taking x2 as the second function

Ex.5 Evaluate dx.

Sol.

(put, 2x + 2 = 3 tanθ ⇒ 2 dx = 3 sec2θ dθ )

Ex.6 If

Sol.

Ex.7 Evaluate

Sol.

Ex.8 Evaluate

Sol.

x where f(x) = tan x = ex f(x) + c = ex tanx + c

Ex.9 Evaluate

Sol.

using, previous example

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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## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

## 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.

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

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