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Evaluate the integral. (Remember the con ∫te ^−4t dt

User Nandsito
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1 Answer

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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.

User Akbaritabar
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