InterviewSolution
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InterviewSolution
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सिद्ध कीजिए कि `int_(0)^(oo)(x)/((1+x)(1+x^(2))dx=(pi)/(4)` |
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Answer» माना `x=tan theta` `rArr" "dx=sec^(2)thetad theta` `{:(x=0,"पर ",tan theta=0, rArr.,theta=0),(x=oo,"पर ",tan theta=oo,rArr.,theta=(pi)/(2)):}` माना `I=int_(0)^(oo)(x)/((1+x)(1+x^(2)))dx` `rArr" "I=int_(0)^(pi//2)(tan theta)/((1+tan theta)(1+tan^(2)theta)).sec^(2) theta d theta` `rArr" "I=int_(0)^(pi//2)(tan theta)/(1+tan theta)d theta" ...(1)"` `rArr" "I=int_(0)^(pi//2)(tan((pi)/(2)-theta))/(1+tan((pi)/(2)-theta))d theta" "` [प्रगुण (4 ) से ] `=int_(0)^(pi//2)(cot theta)/(1+cot theta)d theta=int_(0)^(pi//2)(1//tan theta)/(1+(1)/(tan theta))d theta` `rArr" "I=int_(0)^(pi//2)(1)/(tantheta+1)d theta" ...(2)"` समीकरण (1 ) और (2 ) को जोड़ने पर `2I=int_(0)^(pi//2)(tan theta+1)/(tan theta+1)d theta=int_(0)^(pi//2)d theta=[theta]_(0)^(pi//2)=(pi)/(2)` `rArr" "I=(pi)/(4)" "`यही सिद्ध करना था । |
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