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Question Number 126997 by bramlexs22 last updated on 26/Dec/20

$$\underset{0}{\stackrel{1}{\int }}\arcsin \left(\frac{\sin x}{\sqrt{2}}\right)dx=?$$

Answered by Evimene last updated on 26/Dec/20

$$\mathrm{solution}$$$$\mathrm{let}\sqrt{2}=\alpha$$$$\mathrm{f}\left(\alpha \right)={\int }_0^1\arcsin \left(\frac{\mathrm{sinx}}{\alpha }\right)\mathrm{dx}\iff \mathrm{differentiating}\alpha$$$${\mathrm{f}}^{\prime }\left(\alpha \right)={\int }_0^1\frac{{\alpha }^2}{{\alpha }^2-{\sin }^2\mathrm{x}}\mathrm{dx}\iff \mathrm{multiply}\mathrm{by}\frac{{\sec }^2\mathrm{x}}{{\sec }^2\mathrm{x}}$$$${\mathrm{f}}^{{}^{\prime }}\left(\alpha \right)={\int }_0^1\frac{{\alpha }^2{\sec }^2\mathrm{x}}{{\alpha }^2{\sec }^2\mathrm{x}-{\alpha }^2{\tan }^2\mathrm{x}}\mathrm{dx}\iff \mathrm{recall}{\sec }^2\mathrm{x}+{\tan }^2\mathrm{x}=1$$$${\mathrm{f}}^{\prime }\left(\alpha \right)={\int }_0^1{\sec }^2\mathrm{xdx};\iff {\left[\mathrm{tanx}\right]}_0^1$$$${\mathrm{f}}^1\left(\alpha \right)=\frac{\pi }{4}$$$$\mathrm{f}\left(\alpha \right)=\frac{\pi }{4}\mathrm{x}+\mathrm{c}\iff \mathrm{f}\left(0\right)=0;\mathrm{so}\mathrm{c}=0$$$$\mathrm{f}\left(\alpha \right)=\frac{\pi }{4}$$$$\mathrm{cadet}\mathrm{praise}$$

Commented by bramlexs22 last updated on 26/Dec/20

$$Feynmannmethod?$$

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