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CBSE Class 12-science Answered

Evaluate by substitution and using properties of definite integral
 
 
integral subscript 0 superscript 1 space log left parenthesis square root of 1 plus x end root space space plus space square root of 1 minus x space end root space right parenthesis space d x
Asked by ahuja8087 | 11 Nov, 2016, 06:02: PM
answered-by-expert Expert Answer
begin mathsize 16px style Consider comma space open parentheses square root of 1 plus straight x end root plus square root of 1 minus straight x end root close parentheses squared space equals space 1 plus straight x plus 1 minus straight x minus 2 square root of 1 plus straight x end root square root of 1 minus straight x end root space equals space 2 open parentheses 1 plus square root of 1 minus straight x squared end root close parentheses So comma space log open parentheses square root of 1 plus straight x end root plus square root of 1 minus straight x end root close parentheses space equals log open parentheses 2 open parentheses 1 plus square root of 1 minus straight x squared end root close parentheses close parentheses to the power of 1 divided by 2 end exponent equals 1 half open square brackets log 2 plus log space open parentheses 1 plus square root of 1 minus straight x squared end root close parentheses close square brackets Now comma integral subscript 0 superscript 1 log open parentheses square root of 1 plus straight x end root plus square root of 1 minus straight x end root close parentheses space dx space equals space integral subscript 0 superscript 1 open curly brackets 1 half open square brackets log 2 plus log space open parentheses 1 plus square root of 1 minus straight x squared end root close parentheses close square brackets close curly brackets space dx equals 1 half integral subscript 0 superscript 1 log 2 space dx plus 1 half integral subscript 0 superscript 1 log space open parentheses 1 plus square root of 1 minus straight x squared end root close parentheses space dx end style
 
 
Answered by Rebecca Fernandes | 12 Nov, 2016, 10:31: AM

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