InterviewSolution
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InterviewSolution
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फलन `int(dx)/(5+4 sin x)` का x के सापेक्ष समाकलन कीजिए । |
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Answer» माना `I=int(dx)/(5+4 sinx)` `=int(dx)/(5(sin^(2)x//2+cos^(2)x//2)+8sin x//2 cos x//2)` अंश हर को `cos^(2)x//2` से भाग देने पर , `I=int(sec^(2)x//2dx)/(5(1+tan^(2)x//2)+8tanx//2)` `=(1)/(5)int(sec^(2)x//2dx)/(1+tan^(2)x//2+(8)/(5)tanx//2)` माना `tan x//2 = t therefore sec^(2)x//2 dx=2dt` `therefore" "I=(2)/(5)int(dt)/(1+t^(2)+(8)/(5)t)=(2)/(5)int(dt)/(1+(t^(2)+(8)/(5)t+(16)/(25)-(16)/(25)))` `" "` (पूर्ण वर्ग बनाने पर ) `=(2)/(5)int(dt)/((1-(16)/(25))+(t+(4)/(5))^(2))=(2)/(5)int(dt)/(((3)/(5))^(2)+(t+(4)/(5))^(2))` माना `t+(4)/(5)=u" "rArr" "dt=du" तथा " (3)/(5)=a` `therefore" "I=(2)/(5)int(du)/(a^(2)+u^(2))=(2)/(5a)tan^(-1)((u)/(a))` `=(2)/(5xx(3)/(5))tan^(-1)[(5t+4)/(3)]=(2)/(3)tan^(-1)[(5tan x//2 +4)/(3)]` द्वितीय विधि - `I=int(dx)/(5+4sinx)=int(dx)/(5+(4xx2tanx//2)/(1+tan^(2)x//2))` `=int((1+tan^(2)x//2)dx)/(5(1+tan^(2)x//2)+8 tan x//2)` `=(1)/(5)int(sec^(2)x//2dx)/(1+tan^(2)x//2+(8)/(5)tanx//2)` `=(1)/(5)int(sec^(2)x//2dx)/(1+(tan^(2)x//2+(8)/(5)tanx//2+(16)/(25)-(16)/(25)))` `" "` (पूर्ण वर्ग बनाने पर ) `=(1)/(5)int(sec^(2)x//2dx)/(((3)/(5))^(2)+(tanx//2+(4)/(5))^(2))` माना `" "(3)/(5)=a` तथा `tan x//2+(4)/(5)=u" "therefore" "sec^(2).(x)/(2)dx=2du` `therefore" "I=(2)/(5)int(du)/(a^(2)+u^(2))=(2)/(5a)tan^(-1)((u)/(a))` `=(2)/(3)tan^(-1)[(5tanx//2+4)/(3)]` |
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