Question

How do you solve the antiderivative of ${e^{3x}}$ ?

Answer 1

Here to solve the problem we will use a method of substitution.Finding antiderivatives is nothing but finding the integral. We will substitute the power of e that is $3x = u$ and then taking the derivative we will substitute it in the integral. Finding the integral will give the answer.

Complete step by step solution:
Given that ${e^{3x}}$ is a function so given.
So let substitute $3x = u$
Taking the derivative we get,
$3dx = du$
Now rearrange the terms
$dx = \dfrac{{du}}{3}$
Now coming towards the question we get,
$\int {{e^{3x}}dx}$
Substituting the values so obtained
$\int {{e^{3x}}dx}= \int {{e^u}} \dfrac{{du}}{3}$
Taking constant outside we get,
$\int {{e^{3x}}dx} = \dfrac{1}{3}\int {{e^u}du}$
We know that $\int {{e^x}dx = {e^x} + c}$
Thus we can write,
$\int {{e^{3x}}dx} = \dfrac{1}{3}{e^u} + c$
Resubtituting the used substitution,
$\therefore \int {{e^{3x}}dx}= \dfrac{1}{3}{e^{3x}} + c$

So the antiderivative is $\int {{e^{3x}}dx = \dfrac{{{e^{3x}}}}{3} + c}$ where $c$ is the constant of integral.

Note: We have used the method of substitution because that is quite convenient. We also can use the method in which we can first find the integral of exponential function and then of its power because the type of function also has the order to solve. Also note that we should again substitute the original parameters function because that was a method used to solve the question and not the actual parameters.