what is the integral of e^(-x) , just need verification, ty for answering asap?

Answer 1

-e^(-x)

Any integral e^(b) is e^(b) * the derivative of b.

Answer 2

Integral ( e^(-x) dx )

Advanced Calculus students can do this mentally, but here is how to solve this by hand. Using substitution.

Let u = -x. Then
du = (-1)dx, so
(-1)du = dx

Integral ( e^u (-1) du )

Factor out the (-1) to get

(-1) Integral ( e^u du)

And now, the integral is easy

(-1)e^u + C

back-substitute,

(-1)e^(-x) + C

Answer 3

d/dx e^x = e^x

Proof of e^x : by ln(x)
Given : ln(x) = 1/x; Chain Rule; x = 1.
Solve:

(1) ln(e^x) = x = 1
ln(e^x) = ln(u) e^x (Set u=e^x)
= 1/u e^x = 1/e^x e^x = 1 (equation 1)
e^x = e^x

Discussion of
Integral e^x dx = e^x + C

1. Proof
Since we know the derivative: ex = ex,
we can use the Fundamental Theorem of calculus:
e^x dx = (e^x) dx = e^x + C


since the coefficient of x is -1.

answer is

-e^-x + C

Answer 4

-e^(-x) according to my TI-89.

Answer 5

-e^-x + C