What is the integral of this function?

Question

#int ((e^tan(3x))/(cos^2(3x)))dx #

I'm confused about this problem

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Answer 1 · 2018-04-17T09:44:33

Is an inmediate integral. See below

You know that #d(tanx)/dx=sec^2x=1/cos^2x#

You also know that #inte^(u)u´du=e^(u)+C#

Appliying this, we have

#inte^(tan(3x))/cos^2(3x)dx=1/3e^(tan3x)+C#

Answer 2 · 2018-04-17T09:48:51

=#e^tan(3x)+C#

First
#color(blue)(1/cos^2(3x)=sec^2(3x)#

#inte^tan(3x)/cos^2(3x)dx=inte^tan(3x)sec^2(3x)dx#

#color(blue)(d/dxtan(3x)=sec^2(3x)dx#

#color(blue)(inte^udu=e^u+C##" "# #color(green)(tan(3x)=u" " "in this case"#

Thus

#inte^tan(3x)sec^2(3x)#=#e^tan(3x)+C#