Integral help?

Question

Evaluate the integral or show that it diverges. (Integral of e ^-x dx. with the boundries from 1 to infinity. I am not too good at integrals and I am practicing with various problems I found in a calc book. thank you

Answer 1

Basically what you are doing is called the "integral test" it applies to series.

So If the integral of e^-x is convergent then so is the
sum(1,∞) of e^-x.

The integral of e^-x is -e^-x. This is convergent to 1 as x tends to infinity so the series and integral are convergent.

Answer 2

Int e^-xdx = -e^-x (between +infinity an1)
At+ infinity lim - 1/e^x= 0 and at 1 -e^-x =-1/e
so the integral is 1/e

Answer 3

well, you need to take the limit of the integral as b goes to infinity (replacing the infinity in the integral with b):

b
lim S e^-x dx
b->inf. 0

then, evaluate the integral:

b
lim [-e^-x]
b->inf. 0

Solve:

lim {(-e^-b)-(-e^0)} = lim (-e^b+1)
b->inf. b->inf.

Solve the limit:

-e^(-inf.)+1= 1

The answer is 1