Question #5318

sir please solve"Integral e^x(tanx+1)/secxdx.
best regards

Expert's answer

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**

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

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=

## Comments

Assignment Expert29.11.11, 15:28You are welcome

Akhtar Rasool26.11.11, 09:42thanks honourable expert's you send me the best answer.

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