How To Integrate sqrt[9-x^2]/x^2 by Trigonometric Substitution

Published On :2021-01-28 20:38:00

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begin{array}{l} smallint frac{{sqrt {{bf{9}} - {x^{bf{2}}}} }}{{{x^{bf{2}}}}}{rm{d}}x\\ Let Rightarrow 1 cdot x = 3sin theta ;;;;;;;;;;;;;;\ ;;;;;;;;;;;;;;;{rm{d}}x = 3costheta {rm{d}}theta \ Change{rm{ }}the;expression;;;;;;\ ;;;;;{x^2} = 9{sin ^2}theta ;;;;;;;\ ;;;;9 - {x^2} = 9 - 9{sin ^2}theta ;;;;;;;\ ;;;;9 - {x^2} = 9left( {1 - {{sin }^2}theta } right);;;;;;\ ;;;;;9 - {x^2} = 9{cos ^2}theta ;;;;;;\ ;;;;;sqrt {9 - {x^2}} = 3cos theta \ smallint frac{{3cos theta }}{{9{{sin }^2}theta }} cdot 3cos theta {rm{d}}theta = smallint frac{{co{s^2}theta }}{{si{n^2}theta }}{rm{d}}theta \ smallint {cot ^2}theta {rm{d}}theta = smallint cs{c^2}theta - 1{rm{d}}theta \ Rightarrow - cot theta - theta + c end{array}
begin{array}{l} Rightarrow - frac{{sqrt {9 - {x^2}} }}{x} - {sin ^{ - 1}}frac{x}{2} + c end{array}