#160; ∫ x √(1 x^2) dx x=cos t dx =sin tdt ∫cos t√(1 cos ^2 t)sin t dt ∫cos t dt = sin t+C = sincos ^ 1 x+C = sinsin ^ 1√(1 x^2)+C ...

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Functions

∫ x √1-x2 dx

Question

$\int \frac{x}{\sqrt{1 - x^{2}}} d x$

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Solution

$\int \frac{x}{\sqrt{1 - x^{2}}} d x x = cos t ⟹ d x = sin t d t ⟹ \int \frac{cos t}{\sqrt{1 - {cos}^{2} t}} sin t d t ⟹ \int cos t d t = - sin t + C = - sin {cos}^{- 1} x + C = - sin {sin}^{- 1} \sqrt{1 - x^{2}} + C = - \sqrt{1 - x^{2}} + C$

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