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How do you find the antiderivative of #e^(3x)#?

Answer 1

Paisley Johnson

# int e^(3x) \ dx = 1/3e^(3x) + C#

Using # d/dx e^(ax) = ae^(ax) <=> int ae^(ax) \ dx = e^(ax) + C'#

# :. \ int e^(ax) \ dx = e^(ax)/a + C#

Hence, # int e^(3x) \ dx = 1/3e^(3x) + C#

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Answer 2

Emily Ammerman

To find the antiderivative of ( e^{3x} ), you integrate the function with respect to ( x ):

[ \int e^{3x} , dx ]

Apply the power rule of integration, which states that the antiderivative of ( e^{kx} ) with respect to ( x ) is ( \frac{1}{k} e^{kx} ), where ( k ) is a constant:

[ \int e^{3x} , dx = \frac{1}{3} e^{3x} + C ]

So, the antiderivative of ( e^{3x} ) with respect to ( x ) is ( \frac{1}{3} e^{3x} + 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.