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

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

where does that extra "-2" come from?

To find the antiderivative of the function f(x) = 5e^x - 2sec(x)tan(x), we can follow these steps:

Step 1: Start by finding the antiderivative of 5e^x.
The antiderivative of e^x is e^x itself, so the antiderivative of 5e^x is simply 5e^x.

Step 2: Next, focus on finding the antiderivative of -2sec(x)tan(x).
We need to use a trigonometric identity to simplify this expression. The identity we'll use is:
sec(x)tan(x) = d/dx(sec(x)).
So we can rewrite the expression as -2(d/dx(sec(x))).

Step 3: Now, we need to determine the antiderivative of -2(d/dx(sec(x))).
To find the antiderivative of d/dx(sec(x)), we integrate with respect to x: ∫(d/dx(sec(x)))dx.
This integration will give us the original function: sec(x), so we have -2sec(x).

Step 4: Finally, we can put together the antiderivative of the function f(x).
The antiderivative of 5e^x is 5e^x, and the antiderivative of -2sec(x)tan(x) is -2sec(x).
Thus, the antiderivative of f(x) = 5e^x - 2sec(x)tan(x) is given by: 5e^x - 2sec(x) + C.
Note that C represents the constant of integration and is included because when we take the derivative of the antiderivative, it will vanish.

So, the antiderivative of f(x) = 5e^x - 2sec(x)tan(x) is 5e^x - 2sec(x) + C.