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If \( \int \sin ^{-1} \sqrt{x} d x=x \sin ^{-1} \sqrt{x}+a \sqrt{x} \sqrt{1-x}+b \sin ^{-1} \sqrt{x}+C \) then \( 3(a-b) \) is equal to |
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Answer» If ∫ sin-1 \(\sqrt{x}\) dx = x sin-1 \(\sqrt{x}\) + a\(\sqrt{x}\) \(\sqrt{1-x}\) + b sin-1 \(\sqrt{x}\) + C \(\frac{d}{dx}\) ∫ sin-1 \(\sqrt{x}\) dx = \(\frac{x}{2\sqrt{x}\sqrt{1-x}}+sin^{-1}\sqrt{x}+a\sqrt{x}(\frac{-1}{2\sqrt{1-x}}+\frac{a}{2\sqrt{x}}\sqrt{1-x}+\frac{b}{\sqrt{1-x}}.\frac{1}{2\sqrt{x}}+0)\) ⇒ \(sin^{-1}\sqrt{x}=\frac{\sqrt{x}}{2\sqrt{1-x}}+sin^{-1}\sqrt{x}-\frac{a\sqrt{x}}{2\sqrt{1-x}}+\frac{a\sqrt{1-x}}{2\sqrt{x}}+\frac{b}{2\sqrt{x}\sqrt{1-x}}\) ⇒ \(\frac{x-ax+a(1-x)+b}{2\sqrt{x}\sqrt{1-x}}\) ⇒ x - ax + a - ax + b = 0 ⇒ x(1 - 2a) + (a - b) = 0 = or + 0 ∴ 1 - 2a = 0 and a - b = 0 ⇒ a = 1/2 & b = a = 1/2 ∴ 3(a - b) = 3 x 0 = 0 |
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