InterviewSolution
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InterviewSolution
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Evaluate `int(dx)/(2+sinx+cosx).` |
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Answer» `I=int(dx)/(2+sinx+cosx)` `=int(dx)/(2+(2tanx//2)/(1+tan^(2)x//2)+(1-tan^(2)x//2)/(1+tan^(2)x//2))` `=int(1+tan^(2)x//2)/(tan^(2)x//2+2tanx//2+3)dx` `=int(sec^(2)x//2)/(tan^(2)x//2+2tanx//2+3)dx` `" put " "tan"(x)/(2)=t . :. (1)/(2)"sec"^(2)(x)/(2)dx=dt` `:. I=int(2dt)/(t^(2)+2t+3)` `=2int(dt)/((t+1)^(2)+2)` `=2int(1)/(sqrt(2))"tan"(t+1)/(sqrt(2))+c` `=sqrt(2)"tan"("tan"(x)/(2)+2)/(sqrt(2))+c` |
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