\int_{0}^{1 / 2} {d x}/\sqrt{x-x^{2}} will be equal to

NIMCET 2011 Question Paper with solutions

Question : 36 of 120

Marks: +1, -0

∫↙{0}↖{1 / 2} \;{d x}/{√{x-x^{2}}}

will be equal to

Solution:

∫↙{0}↖{1 / 2} {d x}/{√{x-x^{2}}}=∫↙{0}↖{1 / 2} \;{d x}/{√{x-x^{2}+\;{1}/{4}-\;{1}/{4}}}

=∫↙{0}↖{1 / 2} \;{d x}/{√{ {1}/{4}-(x- {1}/{2})^{2}}}=[\sin ^{-1}( {x-{1}/{2}}/{{1}/{2}})]_{0}^{1 / 2}

=[\sin ^{-1}(2 x-1)]_{0}^{1 / 2}

=\sin ^{-1}(0)-\sin ^{-1}(-1)=0-(- {π}/{2})=\;{π}/{2}

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