решите пожалуйста [tex] \int\ \frac{sin2x}{ \sqrt{4+sin^2x} } \, dx [/tex]. Created by drwnd. matematika-ru

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July 2022 2 32 Report

решите пожалуйста [tex] \int\ \frac{sin2x}{ \sqrt{4+sin^2x} } \, dx [/tex]

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$\int\frac{sin2x}{\sqrt{4+sin^2x}}dx=2\int dt=2t+C=2\sqrt{4+sin^2x}+C\\\\4+sin^2x=t^2\\2tdt=sin2xdx=\ \textgreater \ dx=\frac{2tdt}{sin2x}$

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sedinalana T=√(4+sin²x)⇒t²=4+sin²x
2tdt=2√(4+sin²x)*2*2sinxcosx/2√(4+sin²x)=sin2xdx⇒dx=2tdt/sin2x
∫sin2xdx/√(4+sin²x)=∫sin2x*2tdt/sin2x*t=∫2dt=2t+C=2√(4+sin²x)+C

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