Let `f` beintegrable over `[0,a]` for any real value of `a`. If `I_(1)=int_(0)^(pi//2)cos theta f(sin theta +cos^(2) theta) d theta` and `I_(2)=int_(0

Question

Let `f` beintegrable over `[0,a]` for any real value of `a`. If `I_(1)=int_(0)^(pi//2)cos theta f(sin theta +cos^(2) theta) d theta` and `I_(2)=int_(0

Let `f` beintegrable over `[0,a]` for any real value of `a`.
If `I_(1)=int_(0)^(pi//2)cos theta f(sin theta +cos^(2) theta) d theta` and `I_(2)=int_(0)^(pi//2) sin 2 theta f(sin theta+cos^(2) theta) d theta`, then
A. `I_(1)=-2I_(2)`
B. `I_(1)=I_(2)`
C. `2I_(1)=I_(2)`
D. `I_(1)=-I_(2)`

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Salim Ahmed Kawsar

Correct Answer - B
`I_(1)-I_(2)=int_(0)^(pi//2) (cos theta -sin 2 theta)f(sin theta+cos^(2) theta) d theta`
Put `t=sin theta +cos^(2) theta` and `dt=(cos theta-sin 2 theta)d theta`
or `I_(1)-I_(2)=int_(1)^(1)f(t)dt=0`

4 views Answered 2023-02-05 15:29