How do you find the antiderivative of #e^x(1-(e^-x)sec^2x) dx#?

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Answer 1 · 2017-11-15T22:38:15

#int e^x(1-e^(-x)sec^2x) dx = e^x-tanx+C#

Simplify the expression:

#e^x(1-e^(-x)sec^2x) = e^x -sec^2x#

then using the linearity of the integral:

#int e^x(1-e^(-x)sec^2x) dx = int e^xdx -int sec^2xdx#

Both terms are standard integrals:

#int e^xdx = e^x+c_1#

#int sec^2xdx = tanx +c_2#

and then:

#int e^x(1-e^(-x)sec^2x) dx = e^x-tanx+C#