Question

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

Answer 1

`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`