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How do you find the antiderivative of #e^(-2x)#?

Answer 1

Ezra Adams

Starting from the integral: #inte^(-2x)dx=#

By substitution, you set #-2x=t# so: #x=-t/2# and #dx=-1/2dt#

The integral becomes:

#inte^t(-1/2)dt=-e^t/2+c#

going back to #x#: #=-e^(-2x)/2+c#

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Answer 2

Emily Ammerman

To find the antiderivative of (e^{-2x}), you can use the following steps:

Recognize that (e^{-2x}) represents the exponential function with base (e) raised to the power of (-2x).
Apply the rule for integrating exponential functions: the antiderivative of (e^{kx}) with respect to (x) is (\frac{1}{k}e^{kx} + C), where (C) is the constant of integration.
Substitute (-2x) for (kx) in the integration rule. The antiderivative of (e^{-2x}) becomes (\frac{1}{-2}e^{-2x} + C).
Simplify the expression to get the final antiderivative: (-\frac{1}{2}e^{-2x} + C).

Therefore, the antiderivative of (e^{-2x}) is (-\frac{1}{2}e^{-2x} + C), where (C) is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.