How do you integrate #e^(4x) dx#?

Answer 1

Christopher Agresta

#1/4e^(4x)+C#

We will use the integration rule for #e^x#:

#inte^udu=e^u+C#

So, for the given integral, let #u=4x#. This implies that #du=4dx#.

#inte^(4x)dx=1/4inte^(4x)*4dx=1/4inte^udu=1/4e^u+C#

Since #u=4x#:

#1/4e^u+C=1/4e^(4x)+C#

We can differentiate this answer to check that we get #e^(4x)#. Indeed, through the chain rule, the #1/4# we had to add gets "undone" by the #4# coming from the power of #4x# via the chain rule.

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Answer 2

Luke Alvarez

To integrate ( e^{4x} , dx ), you can use the following method:

Integrate ( e^{4x} , dx ) with respect to ( x ), using the integration rule for ( e^{ax} ), where ( a ) is a constant:

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

Applying this rule with ( a = 4 ), we get:

[ \int e^{4x} , dx = \frac{1}{4} e^{4x} + 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.