Find the antiderivative of f(x)=5e^x-2secxtanx showing steps please

Answer 1

the first term is easy
the derivative of 5e^x is 5e^x
so the integral of 5e^x is 5e^x

now look back at your derivatives of basic trig functions
isn't derivative of secx = secxtanx ?
We have that term
so the integral of secxtanx would be secx
thus the integral of -2secxtanx would simply be -2secx
∫5e^x-2secxtanx = 5e^x - 2 - 2secx

or

simplify -2secx tanx
= -2(1/cosx)(sinx/cosx)
= -2 sinx/cos^2 x
= -2 sinx (cosx)^-2

ahh, I know that derivative of (cosx)^-1
is -1(cosx)^-2 (-sinx) = sinx(cosx)^-2
I have a multiple of that
so integral of -2sinx(cosx)^-2 is -2 (cosx)^-1
or -2/cosx or -2secx

∫5e^x-2secxtanx = 5e^x - 2 - 2secx

Answer 2

where does that extra "-2" come from?