How To Integrate sqrt[x(4-x)] by Trigonometric Substitution

Published On :2021-01-28 20:55:00

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begin{array}{l} smallint sqrt {xleft( {{bf{4}} - x} right)} ;;{rm{d}}x;;;;;;;;;;;;;;;;;,Note;sqrt {a{x^2} + bx + c;} Rightarrow by;Completing;Square\\ Rightarrow smallint sqrt { - left( {{x^2} - 4x + 4 - 4} right)} {rm{d}}x\ Rightarrow smallint sqrt {4 - {{left( {x - 2} right)}^2}} {rm{d}}x;;;;;;;;\ ;;;;;;;;;;;;Let Rightarrow 1 cdot left( {x - 2} right) = 2sin theta ;;;;;;;;;;\ ;;;;;;;;;;;;;;;;;;;;;{rm{d}}x = 2cos theta {rm{d}}theta \ Change{rm{ }}the;expression\ ;;;;;;{left( {x - 2} right)^2} = 4{sin ^2}theta ;;;;;;;\ ;;;;;;4 - {left( {x - 2} right)^2} = 4left( {1 - {{sin }^2}theta } right);;;;;;;\ ;;;;;;4 - {left( {x - 2} right)^2} = 4{cos ^2}theta ;;;;;;;\ ;;;;;sqrt {4 - {{left( {x - 2} right)}^2}} = 2cos theta \ smallint 2cos theta .2cos theta {rm{d}}theta = 4smallint co{s^2}theta ;dtheta \ 4smallint frac{1}{2} + frac{{cos 2theta ;}}{2};;dtheta = 2left( {theta + frac{{sin2theta }}{2}} right) + c\ Rightarrow 2left( {theta + sin theta cos theta } right) + c end{array}
begin{array}{l} Rightarrow 2{sin ^{ - 1}}left( {frac{{x - 2}}{2}} right) + 2 cdot left( {frac{{x - 2}}{2}} right)left( {frac{{sqrt {4 - {{left( {x - 2} right)}^2}} }}{2}} right) + c end{array}