InterviewSolution
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InterviewSolution
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If `int(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)dx=(1)/(12)tan^(-1)(3 tanx)+C`, then the value of ab , isA. `sqrt(41)`B. `sqrt(40)`C. `sqrt(39)`D. `sqrt(38)` |
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Answer» Correct Answer - b We have , `I= int(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)dx` `rArr I= int(sec^(2)x)/(b^(2)+a^(2)tan^(2)x)dx` ` rArr I=(1)/(a)int(1)/(b^(2)+(atanx)^(2))d(atanx)` `=(1)/(ab) tan^(-1)((a)/(b)tanx)+C` `:. ab = 12 and (a)/(b) = 3 rArr a^(2)= 36 rArr a = +-6` `:. ab = 12 rArr b =+-2`. Thus , we have `a sin x +b cos x = +- (6 sin x+2cosx)` We know that `-sqrt(a^(2)+b^(2))le a sin x +b cos x le sqrt(a^(2)+b^(2))` for all x ` :. - sqrt(40)le +- 6 sin x +- 2 cos x le sqrt(40)` for allx. |
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