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calculate-x-1-x-3-2-x-dx-

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Question Number 95691 by mathmax by abdo last updated on 27/May/20
calculate ∫ ((x+1)/( (√((x+3)(2−x)))))dx
calculate∫x+1(x+3)(2−x)dx
Commented by Tony Lin last updated on 27/May/20
∫((x+1)/( (√((x+3)(2−x)))))dx =∫((x+1)/( (√(−x^2 −x+6))))dx =∫((x+(1/2))/( (√(−x^2 −x+6))))dx+∫((1/2)/( (√(−x^2 −x+6))))dx =−(√(−x^2 −x+6))+(1/2)∫(1/( (√(((5/2))^2 −(x−(1/2))^2 ))))dx =−(√(−x^2 −x+6))+(1/2)sin^(−1) ((x−(1/2))/(5/2))+c =−(√(−x^2 −x+6))+(1/2)sin^(−1) ((2x−1)/5)+c
∫x+1(x+3)(2−x)dx=∫x+1−x2−x+6dx=∫x+12−x2−x+6dx+∫12−x2−x+6dx=−−x2−x+6+12∫1(52)2−(x−12)2dx=−−x2−x+6+12sin−1x−1252+c=−−x2−x+6+12sin−12x−15+c

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