Questions: Evaluate the following integral or state that it diverges. [ int1^infty 4 e^-4 x dx ]

Solution Steps

To evaluate the given improper integral, we first find the antiderivative of the integrand \(4 e^{-4x}\). Then, we evaluate the definite integral from 1 to infinity. If the limit exists, the integral converges; otherwise, it diverges.

To evaluate the integral

\[ \int_{1}^{\infty} 4 e^{-4x} \, dx, \]

we first find the antiderivative of the integrand \(4 e^{-4x}\). The antiderivative is given by:

\[ \int 4 e^{-4x} \, dx = -e^{-4x} + C. \]

Next, we evaluate the definite integral from 1 to infinity:

\[ \lim_{b \to \infty} \left[ -e^{-4x} \right]_{1}^{b} = \lim_{b \to \infty} \left( -e^{-4b} + e^{-4} \right). \]

As \(b\) approaches infinity, \(e^{-4b}\) approaches 0. Therefore, we have:

\[ \lim_{b \to \infty} \left( -e^{-4b} + e^{-4} \right) = 0 + e^{-4} = e^{-4}. \]

Since the limit exists and is finite, the integral converges.

Final Answer