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Evaluate:`int_0^1(2-x^2)/((1+x)sqrt(1-x^2))dx` |
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Answer» Correct Answer - `(pi)/2` Put `x=sintheta`. So `dx=cos theta d theta` `:.` Given integral `=int_(0)^(pi//2)((2-sin^(2)theta)cos theta d theta)/((1+sin theta) cos theta)` `=int_(0)^(pi//2)(1-sin theta+1/(1+sin theta))d theta` `=|theta + cos theta|_(0)^(pi//2)+int_(0)^(pi//2)(d theta)/(1+sin theta)` `=(pi)/2-1+int_(0)^(pi//2)(1-sin theta)/(cos^(2)theta) d theta` `(pi)/2-1+int_(0)^(pi//2) (sec^(2) theta -sec theta tan )d theta` `=(pi)/2-1+|tan theta -sec theta|_(0)^(pi//2)` `=(pi)/2-1+lim_(theta to pi//2)(sin theta-1)/(cos theta)-(sin 0-1)/(cos 0)` `=(pi)/2-1+lim_(theta to pi//2) (cos theta)/(sin theta) +1` `=(pi)/2` |
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