How do you evaluate the integral #int e^(5x)#?

Question

Scarlett Allison · Accepted Answer

I got:
#e^(5x)/5+c# Here you need to find a function (Primitive) that derived gives you #e^(5x)#. We can guess immediately that this function could be: #e^(5x)/5+"constant"# that derived gives us the integrand; or we can use the general rule to say that: #inte^(kx)dx=e^(kx)/k+c# where #k and c# are two constants. We need the second constant #c# in the result to cover all the possibilities for our Primitive Function.

Emily Ammerman · Answer

undefined To evaluate the integral $ \int e^{5x} $, you can use the following steps:

1. Recognize that $ \int e^{5x} $ is a simple exponential function.
2. Apply the power rule for integration, which states that $ \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C $, where $ a $ is a constant and $ C $ is the constant of integration.
3. Using the power rule, for $ \int e^{5x} $, the constant $ a $ is $ 5 $, so the integral becomes $ \frac{1}{5} e^{5x} + C $.
4. Add the constant of integration $ C $ to represent the family of antiderivatives.

Therefore, $ \int e^{5x} \, dx = \frac{1}{5} e^{5x} + C $.