Integration By Parts

%5Cint%20%5E%7B1%2F2%7D_%7B-1%2F2%7D%20%5Cbigg%20(%5Bx%5D%20%2B%20log%20%5Cbigg%20(%20%5Cfrac%7B1%2B%20x%7D%7B1%20-%20x%7D%20%5Cbigg%20)%20%5Cbigg%20)%20%5C%2C%20dx%20%5C%2C

%3D%20%5Cint%5E%7B1%2F2%7D_%7B-1%2F2%7D%20%5Bx%5D%20%5C%2C%20dx%20%2B%20%5Cint%5E%7B1%2F2%7D_%7B-1%2F2%7D%20%5C%2C%20log%20%5Cbigg%20(%5Cfrac%7B1%2B%20x%20%7D%7B1%20-%20x%20%7D%20%5Cbigg%20)%20dx

%3D%20%5Cint%5E%7B1%2F2%7D_%7B-1%2F2%7D%20%5Bx%5D%20%5C%2C%20dx%20%2B%200%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5Cbigg%20%5B%5Cbecause%20%5Cbigg%20(%20%5Cfrac%7B%201%20%2B%20x%20%7D%7B%201%20-%20x%7D%20%5Cbigg%20)

is an odd function

%5Cbigg%20%5D

%3D%20%5Cint%5E%7B0%7D_%7B-1%2F2%7D%20%5Bx%5D%20%5C%2C%20dx%20%2B%20%5Cint%5E%7B1%2F2%7D_%7B0%7D%20%5Bx%5D%20%5C%2C%20dx%20%3D%20%5Cint%5E%7B0%7D_%7B-1%2F2%7D%20(%20-%201)%20%5C%2C%20dx%20%2B%20%5Cint%5E%7B1%2F2%7D_%7B0%7D%20%5C%2C%20dx

%3D%5B%20x%5D%5E0_%7B-1%2F2%7D%20%3D%20-%20%5Cbigg%20(%200%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20%5Cbigg%20)%20%3D-%20%5Cfrac%7B1%7D%7B2%7D

The correct option is:(A) $%5Cfrac%7B2%5En%7D%7Bm%2B1%7D-%5Cfrac%7Bn%7D%7Bm%2B1%7D(m%2B1%2Cn-1)$ .

Here, I(m,n) = $%5Cint_0%5E1%20t%5Em(1%2Bt)n%20dt%20%2C$ reduce into I(m + 1, n - 1)
[we apply integration by parts taking (1 + t) $%5En$ as first
and t $%5Em$ as second function]
$%5Ctherefore%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20i(m%2Cn)%3D%5Cbigg%5B(1%2Bt)%5En.%5Cfrac%7Bt%5E%7Bm%2B1%7D%7D%7Bm%2B1%7D%5Cbigg%5D_0%5E1%20%5C%2C%20-%5Cint_0%5E1n(1%2Bt)%5E%7Bn-1%7D.%5Cfrac%7Bt%5E%7Bm%2B1%7D%7D%7Bm%2B1%7Ddt$
$%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%3D%5Cfrac%7B2%5En%7D%7Bm%2B1%7D-%5Cfrac%7Bn%7D%7Bm%2B1%7D%20%5Cint_0%5E1%20(1%2Bt)%5E%7B(n-1)%7D%2Ct%5E%7Bm%2B1%7Ddt$
$%5Ctherefore%20%5C%2C%20%5C%2C%20I(m%2Cn)%20%3D%5Cfrac%7B2%5En%7D%7Bm%2B1%7D-%5Cfrac%7Bn%7D%7Bm%2B1%7D.I(m%2B1%2Cn-1)$

About Integration By Parts - JEE-ADVANCED

Integration By Parts is a vital chapter for JEE-ADVANCED aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

Why focus on Integration By Parts PYQs?

Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Integration By Parts carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

About Integration By Parts - JEE-ADVANCED

Frequently Asked Questions

Why focus on Integration By Parts PYQs?

How to best use this analysis?

Integration By Parts

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

About Integration By Parts - JEE-ADVANCED

Frequently Asked Questions

Why focus on Integration By Parts PYQs?

How to best use this analysis?

Chapter Questions
2 MCQs