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Question Number 125837 by joki last updated on 14/Dec/20

p r o v e t h a t :

\int \sqrt{a^{2} - x^{2}} d x =_{} \frac{x}{2} \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} {s i n}^{- 1} (\frac{x}{a}) + c

Answered by Dwaipayan Shikari last updated on 14/Dec/20

T a k e x = a s i n θ, 1 = a c o s θ \frac{d θ}{d x}

\int \sqrt{a^{2} - x^{2}} d x = \int a c o s θ \sqrt{a^{2} - a^{2} {s i n}^{2} θ} d θ

= a^{2} \int {c o s}^{2} θ d θ = \frac{a^{2}}{2} \int 1 + c o s 2 θ d θ = \frac{a^{2} θ}{2} + \frac{a^{2}}{4} s i n 2 θ + C

θ = {s i n}^{- 1} \frac{x}{a} s i n 2 θ = 2 \frac{x}{a^{2}} \sqrt{a - x^{2}}

S o \int \sqrt{a^{2} - x^{2}} d x = \frac{a^{2} {s i n}^{- 1} \frac{x}{a}}{2} + \frac{x}{2} \sqrt{a^{2} - x^{2}} + C