##3e^x+C##

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##

##3e^x+C##

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##