sir please solve"Integral e^x(tanx+1)/secxdx.
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Expert's answer
2011-11-25T08:31:53-0500
it's known that secx=1/cos(x), tan x=sinx/cosx hence (tanx+1)/secx=(tanx+1)cosx=(sinx+cosx) So Integral(e^x(tanx+1)/secx)dx=Integral(e^x*(sinx+cosx))dx=Integral(e^x*sinx)dx+Integral(e^x*cosx)dx (*) Consider the first one: Integral(e^x*sinx)dx = e^x*sinx-Integral(e^x*cosx)dx +const (**) using integration by parts u=sinx => u'=cosx v'=e^x => v=e^x Substituting obtained result (**) into formula for initial integral (*) we get Integral(e^x(tanx+1)/secx)dx=e^x*sinx-Integral(e^x*cosx)dx +const+Integral(e^x*cosx)dx=e^x*sinx+const
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