Question

evaluate the improper integral or state that it diverges.
integral of dx/(e^x + e^-x) from negative infinity to infinity.

Answer 1

\[\int\limits_{-\infty}^{\infty} \frac{ dx }{ e^{x}+e ^{-x} }\]

Answer 2

I think it would help to get rid of that negative exponent by multiplying both top and bottom by e^x
and then use a trig sub

Answer 3

@myininaya i dont understand what you mean by using a trig substitution. doesnt that involve sin and cos not e??

Answer 4

\[\int\limits \frac{e^x}{e^{2x} +1}dx\]
since
\[\tan^2 +1 = \sec^2\]
let
\[e^x = \tan u\]
\[e^x dx = \sec^2 u du\]
\[\rightarrow \int\limits \frac{\sec^2 u}{\sec^2 u} du = \int\limits du = u + C\]

Answer 5

oops you dont need "+C" since you're evaluating at infinity

Answer 6

but i need to find whether it diverges or converges using limits..

Answer 7

right
\[\lim_{x \rightarrow \infty} \tan^{-1} (e^x) - \lim_{x \rightarrow -\infty} \tan^{-1} (e^x)\]