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:

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

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