InterviewSolution
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InterviewSolution
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`int sqrt(tanx)dx` का मान ज्ञात कीजिए । |
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Answer» माना `I=intsqrt(tanx)dx` `=int sqrt(t^(2)).(2tdt)/(1+t^(4))" "{:("माना "tanx=t^(2)),(therefore sec^(2)xdx=dt.2t):}` `=int((t^(2)+1)+(t^(2)-1))/(t^(4)+1)dt" "rArr dx=(2t dt)/(1+tan^(2)x)` `=int(t^(2)+1)/(t^(4)+1)dt+int(t^(2)-1)/(t^(4)+1)dt" "=(2tdt)/(1+t^(4))` `=int(1+(1)/(t^(2)))/(t^(2)+(1)/(t^(2)))dt+int(1-(1)/(t^(2)))/(t^(2)+(1)/(t^(2)))dt` `=int(1+(1)/(t^(2)))/((t-(1)/(t))^(2)+(sqrt2)^(2))dt+int(1-(1)/(t^(2)))/((t+(1)/(t))^(2)-(sqrt2)^(2))dt` माना प्रथम समाकल के लिये `t-(1)/(t)=u rArr (1+(t)/(t^(2)))dt=du` दूसरे समाकल के लिये `t+(1)/(t)= v rArr (1-(1)/(t^(2)))dt=dv` `therefore" "I=int(1)/(u^(2)+(sqrt2)^(2))du+int(1)/(v^(2)-(sqrt2)^(2))dv` `=(1)/(sqrt2)tan^(-1)((u)/(sqrt2))+(1)/(2sqrt2)log|(v-sqrt2)/(v+sqrt2)|+c` |
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