How do you integrate $\int {{e^{3x}}dx} $?

Hint: First of all we will convert the given expression in the simplified form and then place the relevant formulas and simplify for the resultant required values. To simplify the expression, take any variable as the reference term and replace it in the given expression and also use differentiation concepts to get it and then simplify for the required solution. Here we will take 3x equal to some variable and solve accordingly.

Complete step-by-step answer:
Take the given expression: $\int {{e^{3x}}dx} $ …. (A)
Here, we will apply the formula for integration by substitution.
Let us assume $u = 3x$ ---(B)
Take differentiation on both the sides with respect to “x”
$\dfrac{{du}}{{dx}} = \dfrac{{3x}}{{dx}}$
Take constant outside on the right hand side of the equation –
$\dfrac{{du}}{{dx}} = \dfrac{{3(x)}}{{dx}}$
Now, $\dfrac{x}{{dx}} = 1$ place it in the above expression –
$\dfrac{{du}}{{dx}} = 3(1)$
Simplify the above expression –
$\dfrac{{du}}{{dx}} = 3$
Perform cross multiplication where the denominator of one side is multiplied with the numerator of the opposite side and vice-versa and make the required term the subject –
$dx = \dfrac{{du}}{3}$ -- (c)
Now, place the values of equation (B) and (C) in the equation (A)
$\int {{e^{3x}}dx = \int {\dfrac{1}{3}{e^u}} du} $
Also, use $\int {{e^x}dx = {e^x} + C} $
$\int {{e^{3x}}dx = \dfrac{1}{3}{e^{3x}} + C} $
This is the required solution.
So, the correct answer is “$\dfrac{1}{3}{e^{3x} + C}$”.

Note: Do not get confused between the term differentiation and the integration. Both are inverse of each other. Integration is the concept of calculus and it is the act of finding the integrals whereas differentiation can be represented as the rate of change of the function. Always remember the above example can be solved by the chain rule method, where the integration takes place continuously till the power of “x” comes one.