InterviewSolution
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InterviewSolution
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`int_0^a(dx)/(sqrt(a x-x^2))` |
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Answer» `(i) int_(0)^(a)(dx)/(sqrt(ax-x^(2)))=int_(0)^(a)(dx)/(sqrt(-(x^(2)-ax+(a^(2))/(4))+(a^(2))/(4)))` `=int_(0)^(a)(dx)/(sqrt(((a)/(2))^(2)-(x-(a)/(2))^(2)))` `=[sin^(-1).((x-(a)/(2)))/(((a)/(2)))]_(0)^(a)=[sin^(-1)((2x-a)/(a))]_(0)^(a)` `=[sin^(-1)(1)-sin^(-1)(-1)]` `=2sin^(-1)(1)=(2xx(pi)/(2))=pi`. `(ii) int_(0)^(sqrt(2))sqrt(2-x^(2))dx=int_(0)^(sqrt(2))sqrt((sqrt(2))^(2)-x^(2))dx` `=[(x)/(2)sqrt(2-x^(2))+((sqrt(2))^(2))/(2)*sin^(-1).(x)/(sqrt(2))]_(0)^(sqrt(2))` `=[0+sin^(-1)(1)]-[0+sin^(-1)0]=(pi)/(2)`. |
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