Final answer:
The integral of e^-4t dt is evaluated using the fundamental rules of integration, yielding a result of -(1/4)e^-4t + C.
Step-by-step explanation:
To evaluate the integral of e−4t dt, you can apply the basic rules of integration.
The integral of eax dx is (1/a)eax + C, where C is the constant of integration. Therefore, the integral of e−4t dt with respect to t is −(1/4)e−4t + C.
Certainly! Let's break down the process of finding the integral step by step:
Given integral:
\[ \int e^{-4t} \,dt \]
To find the antiderivative (indefinite integral), we'll use the power rule for integration. The power rule states that if you have an integral of the form \(\int a \cdot e^{kx} \,dx\), the antiderivative is \(\frac{a}{k}e^{kx} + C\), where \(C\) is the constant of integration.
In your case, the integral is:
\[ \int e^{-4t} \,dt \]
To apply the power rule, let's identify \(a\), \(k\), and then use the formula:
\[ a = 1 \]
\[ k = -4 \]
Now apply the power rule:
\[ \int e^{-4t} \,dt = -\frac{1}{4}e^{-4t} + C \]
Here, \(C\) is the constant of integration.
So, the indefinite integral of \(e^{-4t}\) with respect to \(t\) is \(-\frac{1}{4}e^{-4t} + C\).
If you have specific limits of integration (a definite integral), please provide them so that we can further evaluate the integral.