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]
$%5Ctherefore%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20i(m%2Cn)%3D%5Cbigg%5B(1%2Bt)%5En.%5Cfrac%7Bt%5E%7Bm%2B1%7D%7D%7Bm%2B1%7D%5Cbigg%5D_0%5E1%20%5C%2C%20-%5Cint_0%5E1n(1%2Bt)%5E%7Bn-1%7D.%5Cfrac%7Bt%5E%7Bm%2B1%7D%7D%7Bm%2B1%7Ddt$
$%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%3D%5Cfrac%7B2%5En%7D%7Bm%2B1%7D-%5Cfrac%7Bn%7D%7Bm%2B1%7D%20%5Cint_0%5E1%20(1%2Bt)%5E%7B(n-1)%7D%2Ct%5E%7Bm%2B1%7Ddt$
$%5Ctherefore%20%5C%2C%20%5C%2C%20I(m%2Cn)%20%3D%5Cfrac%7B2%5En%7D%7Bm%2B1%7D-%5Cfrac%7Bn%7D%7Bm%2B1%7D.I(m%2B1%2Cn-1)$

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

Integration By Parts

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

Chapter Questions
2 MCQs