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How do I find the antiderivative of #f(x)=e^(-5x)#?

Answer 1

#inte^(-5x)dx=-1/5e^(-5x)+C#

In general, the antiderivative of an exponential in the form

#inte^(ax)dx=1/ae^(ax)+C# where #a# is some non-zero constant.

This makes sense -- were we to differentiate #1/ae^(ax)+C,# we'd get #a/ae^(ax)=e^(ax)#, making this a suitable antiderivative.

Thus,

#inte^(-5x)dx=-1/5e^(-5x)+C#

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Answer 2

Scarlett Allison

To find the antiderivative of ( f(x) = e^{-5x} ), you can use the power rule for integration:

[ \int e^{ax} , dx = \frac{1}{a} e^{ax} + C ]

Using this rule, where ( a = -5 ), the antiderivative of ( e^{-5x} ) is:

[ \int e^{-5x} , dx = \frac{1}{-5} e^{-5x} + C = -\frac{1}{5} e^{-5x} + C ]

Where ( C ) is the constant of integration. So, the antiderivative of ( f(x) = e^{-5x} ) is ( -\frac{1}{5} e^{-5x} + C ).

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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.