Expert answers to "Integrate ∫01sin−1{x21+x2}dx"

Turnseeuw

Answered question

2022-01-30

Integrate

\int_{0}^{1} \sin^{- 1} {\frac{x^{2}}{1 + x^{2}}} d x

egowaffle26ic

Beginner2022-01-31Added 7 answers

\int_{0}^{1} \arcsin (\frac{x^{2}}{1 + x^{2}}), d x = \int_{0}^{1} \arcsin (\frac{x}{1 + x}) \frac{d x}{2 \sqrt{x}}

equals

\int_{0}^{\frac{1}{2}} \frac{\arcsin (u)}{2 u^{\frac{1}{2}} {(1 - u)}^{\frac{3}{2}}} d u

which can be managed by integration by parts. It boils down to

\frac{π}{6} - \int_{0}^{\frac{1}{2}} \sqrt{\frac{u}{1 - u}} \cdot \frac{d u}{\sqrt{1 - u^{2}}} d u

then to

\frac{π}{6} + \log (2 + \sqrt{3}) - \sqrt{2} \log (\sqrt{2} + \sqrt{3}) \approx 0.219563

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