1. ## Improper Integrals!!!!!!

Hi! The problem is: determine whether or not the integral is convergent or divergent, and evaluate if convergent: The integral from negative infinity to negative one of e^-4x dx. I thought that the answer might be zero, because the graph approaches zero, but that might be positive infinity. Please help, I'm really not getting improper integrals!! And also, how do you know if an integral is improper?

2. Originally Posted by elocin
Hi! The problem is: determine whether or not the integral is convergent or divergent, and evaluate if convergent: The integral from negative infinity to negative one of e^-4x dx. I thought that the answer might be zero, because the graph approaches zero, but that might be positive infinity. Please help, I'm really not getting improper integrals!! And also, how do you know if an integral is improper?
An integral is improper when:

1) One/Both of the limits contain $\infty$. Ex: $\int_{-\infty}^{\infty}e^{-x^2}\,dx$

2) When one of the limits exists at a undefined value of a function. Ex: $\int_0^5\frac{1}{x}\,dx$

3) When the integral is being over an interval in which an asymptotic discontinuity occurs. Ex: $\int_{-7}^{5}\frac{4}{x-3}\,dx$

$\int_{-\infty}^{-1}e^{-4x}\,dx=\left.\left[-\frac{1}{4}e^{-4x}\right]\right|_{-\infty}^{-1}=-\frac{1}{4}e^4+\frac{1}{4}e^{\infty}=\infty$