Question

What is the antiderivative of ${e^{2x}}$?

Answer 1

Antiderivative is the opposite of derivative. It is the same as integrals or integration. In order to find the antiderivative of ${e^{2x}}$, substitute the value of $2x = u$ in ${e^{2x}}$,then integrate according to the best suitable methods of integration known to us. And the best method that would be suitable for this function would be integration by substitution, that is replacing a function with a new variable, then integrating the function according to that variable only.

Formula used:
$\int {{e^y}dy = {e^y} + C}$

Complete step by step answer:
The antiderivative of the exponential ${e^{2x}}$.
Substitute $2x = u$ in the exponential ${e^{2x}}$, and we get: ${e^u}$.
Differentiating $2x = u$ with respect to $x$, we get:
$2x = u \\\ \Rightarrow \dfrac{{d2x}}{{dx}} = \dfrac{{du}}{{dx}} \\\ \Rightarrow 2\dfrac{{dx}}{{dx}} = \dfrac{{du}}{{dx}} \\\ \Rightarrow 2dx = du \\\ \Rightarrow dx = \dfrac{{du}}{2} \\\$
Integrating ${e^{2x}}$ with respect to $dx$;
$\int {{e^{2x}}} dx$
Substituting the value of $dx = \dfrac{{du}}{2}$ and $2x = u$ in the exponential ${e^{2x}}$ in the above function, and we get:
$\int {{e^u}} \dfrac{{du}}{2}$
As, we know that $\int {{e^y}dy = {e^y} + C}$
So using this formula integrating the above function, we get:
$\dfrac{1}{2}\int {{e^u}} du = \dfrac{{{e^u}}}{2} + C$
Now, substituting back the original values of $u = 2x$ in the above resultant and we get:
$\dfrac{{{e^{2x}}}}{2} + C$. We can also cross check the result by differentiating the output with respect to $x$.

Therefore, the antiderivative of ${e^{2x}}$ is $\dfrac{{{e^{2x}}}}{2} + C$.

Note: It’s very important to add a constant after integrating the equation because when we derive an equation their constant is removed. For example, let’s see an equation: $2x + 2$. Derivative this with respect to $x$ and we get: $\dfrac{{d(2x + 2)}}{{dx}} = \dfrac{{d2x}}{{dx}} + \dfrac{{d2}}{{dx}} = 2 + 0 = 2$.Because we know that derivation of constant term is zero. Now Let’s check and integrate the answer obtained and we get: $\int {2dx} = 2x$ and we can see that the constant term is lost. That’s why we need to add a constant term at last after integration.