What is the antiderivative of #3e^x#?

1 Answer
Jan 27, 2016

#3e^x+C#

Explanation:

You should already know that the derivative of #e^x# is just #e^x#. Also, when differentiating, multiplicative constants remain and are not altered.

Since the two components of this function are a multiplicative constant #3# and #e^x#, we can say that #d/dx(3e^x)=3e^x#.

Thus, the antiderivative of the function is just #3e^x+C#.

The #C#, or the constant of integration, is added because constants have no bearing when finding a derivative.

More formally, we could use substitution.

#{(u=x),((du)/dx=1=>du=dx):}#

We want to find

#int3e^xdx=3inte^xdx#

Simplify with #u# substitution:

#=3inte^udu#

Use the rule that #inte^udu=e^u+C#

#=3e^u+C=3e^x+C#